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Preview — Castle Dor by Daphne du Maurier
Castle Dor
Daphne du Maurier,
Arthur Quiller-Couch
Nina Bawden (Introduction)
Both a spellbinding love story and a superb evocation of Cornwall's mythic past, Castle Dor is a book with unique and fascinating origins.
It began life as the unfinished last novel of Sir Arthur Quiller-Couch, the celebrated 'Q', and was passed by his daughter to Daphne du Maurier whose storytelling skills were perfectly suited to the task of completing the old master's t Both a spellbinding love story and a superb evocation of Cornwall's mythic past, Castle Dor is a book with unique and fascinating origins.
It began life as the unfinished last novel of Sir Arthur Quiller-Couch, the celebrated 'Q', and was passed by his daughter to Daphne du Maurier whose storytelling skills were perfectly suited to the task of completing the old master's tale.
The result is this magical, compelling recreation of the legend of Tristan and Iseult, transplanted in time to the Cornwall of the last century. A chance encounter between the Breton onion-seller, Amyot Trestane, and the newly-wed Linnet Lewarne launches their tragic story, taking them in the fateful footsteps of the doomed lovers of Cornish legend ... ...more
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Published June 3rd 2004 by Virago (first published 1961)
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Feb 15, 2014 Sarah (Presto agitato) rated it it was ok · review of another edition
Shelves: historical-fiction, awaken-the-spirits
The legend of Tristan, the Cornish knight, and his doomed love for Iseult, the wife of King Mark, is an ancient one, appearing in many variations for hundreds of years. Castle Dor is a retelling of that story set in Cornwall in the 1840s. It was begun by Arthur Quiller-Couch, a British novelist who wrote under the pen name "Q," but left unfinished mid-chapter. Many years after his death, his daughter asked Daphne du Maurier to complete it.
Daphne du Maurier, queen of the Gothic novel, would seem The legend of Tristan, the Cornish knight, and his doomed love for Iseult, the wife of King Mark, is an ancient one, appearing in many variations for hundreds of years. Castle Dor is a retelling of that story set in Cornwall in the 1840s. It was begun by Arthur Quiller-Couch, a British novelist who wrote under the pen name "Q," but left unfinished mid-chapter. Many years after his death, his daughter asked Daphne du Maurier to complete it.
Daphne du Maurier, queen of the Gothic novel, would seem like the perfect person to take on this tale. Star-crossed lovers, betrayal, the dark and rocky Cornish coast, hints of the supernatural - it's a story that is right in her wheelhouse.
Unfortunately, the book never recovers from Q's beginning. We don't know exactly where Quiller-Couch stopped writing and du Maurier began, but I feel certain she had no part in this description:
"This most ancient cirque of Castle Dor, deserted, bramble-grown, was the very nipple of a huge breast in pain, aching for discharge."
The pages and pages of florid description and stilted conversation, punctuated by an occasional ill-considered metaphor and painfully obvious info dumping, are mind-numbing after a while.
Maybe a little more than a third of the way through, things get better. This line, which refers to the legend of Tristan and Iseult, seems to be a sly hint that there is someone new steering the ship:
"It is a curious coincidence that no poet, or shall we call him investigator, has ever lived to conclude this particular story. His work has always been finished by another."
The change in style is gradual. By the end of the novel it is a much more enjoyable book, but it never overcomes its beginning. The character's motivations are far-fetched. If she had set it up from the start, I think du Maurier might have pulled off the (view spoiler)[vaguely supernatural (hide spoiler)] explanation for their actions, but it never really works. Daphne du Maurier wrote many better books than this one. It may be worth reading for dedicated du Maurier fans, but most should probably give it a miss. ...more
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Sep 28, 2012 Nikki rated it liked it
Shelves: classics, based-on-myth-saga-etc
A rather less read du Maurier book -- fascinating, seeing her continuing someone else's work. And I agree with the introduction that it's hard to tell where she picked up the story: there's a shift somewhere, I think, in the tone of the beginning and the tone of the end, but it all flows smoothly enough.
I can't really give it four stars in terms of enjoyment, because I thought some of the parallels with the Tristan and Iseult story were overlaboured, and all the details of geography meant little A rather less read du Maurier book -- fascinating, seeing her continuing someone else's work. And I agree with the introduction that it's hard to tell where she picked up the story: there's a shift somewhere, I think, in the tone of the beginning and the tone of the end, but it all flows smoothly enough.
I can't really give it four stars in terms of enjoyment, because I thought some of the parallels with the Tristan and Iseult story were overlaboured, and all the details of geography meant little to me (you'd think it would be more interesting to me, given my research into the Arthurian legends, but actually I have very little interest in whether they're fact or fiction) -- it didn't add to the story, for me, because I didn't need to think that Linnet was somehow descended from La Belle Iseut's family or that Mary might be descended from Isolde of the White Hands. I quite like the replaying-old-stories trope, and to me it's closeness of feeling and mindset that works best to connect the characters, not blood kinship diluted over hundreds of years.
Anyway, despite that, I actually found it more of a page turner than most of du Maurier's other works. I'd open it up and look up what I thought was five minutes later to discover that an hour had somehow passed. I can't put my finger on exactly why this was, but I enjoyed the book, anyway. ...more
Oct 04, 2010 Hannah rated it it was ok · review of another edition
Shelves: 2011-reads, historicals, romances
Set in 1860's Cronwall, Castle Dor weaves the boring tale of Amyot (a young Breton sailor) and Linnet (the young and beautiful wife of an extremely old man). How the pair meet, love and ultimately live out the tragic end of their fabled counterparts (Tristan & Iseult) makes for a snooze fest of epic proportions.
Hard to imagine Daphne duMaurier co-authored this book. IMO, Arthur Quiller-Couch wrote the majority of it, since it was extremely dry and esoteric, and I know for a fact that duMau Set in 1860's Cronwall, Castle Dor weaves the boring tale of Amyot (a young Breton sailor) and Linnet (the young and beautiful wife of an extremely old man). How the pair meet, love and ultimately live out the tragic end of their fabled counterparts (Tristan & Iseult) makes for a snooze fest of epic proportions.
Hard to imagine Daphne duMaurier co-authored this book. IMO, Arthur Quiller-Couch wrote the majority of it, since it was extremely dry and esoteric, and I know for a fact that duMaurier's writing style is anything but. How anyone could make a retelling of the story of Tristan and Iseult boring is beyond me, but alas, it happened here for me.
Highly recommended for readers who need to fall asleep, but don't want the effects of medication to make that happen.
2 stars because the writing (in some parts) is pretty lovely and lyrical. Must have been Daphne's lines... ...more
Nov 16, 2015 Stephanie Davies rated it liked it · review of another edition
I'm not going to do a proper review, I just want to share the first description of the titular castle that you see in the book:
Feb 19, 2014 Tinka rated it did not like it · review of another edition
Shelves: paperback-hardcover, stand-alone, read-in-german, classics, 1-star-you-suck, dnf
"She's dearer than life itself, that's all I know"
WTF did I just read?
Like seriously, I have no idea. I started this book and the prologue was weird, but I thought 'hey, it's just the prologue…' haha silly me thinking it would get better. Nope, it only got worse.
I picked this book believing it was only written by Daphne Du Maurier. I'm not that into the Tristan and Isolde myth, but I thought if Daphne Du Maurier with her haunting storytelling takes on that legend, it can only be good and atmos "She's dearer than life itself, that's all I know"
I picked this book believing it was only written by Daphne Du Maurier. I'm not that into the Tristan and Isolde myth, but I thought if Daphne Du Maurier with her haunting storytelling takes on that legend, it can only be good and atmospheric. What my book edition didn't tell me however is the fact that this book is not only written by Daphne, but majorly by Sir Arthur Quiller-Crouch. It was originally his story and writing, sadly he died before he could finish it and it was his daughter's wish for Du Maurier to finish it, which she did. There are traces of Daphne's usual writing style in this book, but it is mostly blink and you miss it. Most of the writing seems still to be Quiller-Crouch's (or Daphne did a fabulous job writing like him, I don't know I have never read anything by him) and let me tell you it is just not mine.
The writing seems more like a scientific report than an actual novel, it is artificial, gives you only rough descriptions of everything that is going on, including characters and setting, there is no atmosphere at all and for a retelling of a tragic love story one important thing is missing, the actual love.
There is no build up, no charm, no passion, no feelings. There are just two characters or rather "names" (because for an actual character I demand a personality) and they fall in love and it is forbidden because she is married. Also there are a bunch of other names.
The author('s) reference the original tale of Tristan and Isolde a lot and to me that seemed weird. Meta can be good, but in a story that is promoted as an actual 'modern' retelling, the Meta aspects felt highly out of place.
There is a lot of esoteric stuff in the book as well which is simply not my cup of tea.
I hardly ever stop reading a book before finishing it, but I gave up on this one about 150 pages into the story. There was nothing that kept my interest for the slightest and continuing reading it would seem like a waste of time.
Conclusion: Love story without love. Odd writing style. Just not mine.
Recommendation: I wouldn't even recommend it to hardcore Daphne Du Maurier fans, because it hardly seemed to be her writing at all. The style is very different, I mean I think it could be good for some people, but I just don't get it. ...more
Mar 09, 2009 Misfit rated it liked it
"Not in your world.....but in some borderland of buried kings and lovers"
Linnette Lewarne, married to a much older man, meets Breton Amyot by pure chance and their fates are forever sealed as they begin to relive a past that has happened time and time again through the centuries - that of Tristan and Iseult. Doctor Carfax watches from the sidelines as he puts the pieces of the puzzle together with that of the legends and ends with a race against time to stop the legend from repeating itself into "Not in your world.....but in some borderland of buried kings and lovers"
Linnette Lewarne, married to a much older man, meets Breton Amyot by pure chance and their fates are forever sealed as they begin to relive a past that has happened time and time again through the centuries - that of Tristan and Iseult. Doctor Carfax watches from the sidelines as he puts the pieces of the puzzle together with that of the legends and ends with a race against time to stop the legend from repeating itself into tragedy once again - all culminating in a on a very foggy Cornwall All Hallows E'en. Is the good Doctor in time or not?
Well you know me, I don't tell. Castle Dor, unfinished at the death of author Sir Arthur Quiller-Couch ("Q"), was completed by Du Maurier at the request of his daughter. A bit slow and dry at the start (I've not read anything from "Q" before, nor am I all that familiar with the legends of Tristan and Iseult), but a good finish, albeit not the strongest. If you're big into the legends of T&I I'd go for it, but Du Maurier fans will probably be disappointed - the parts she contributed at the end are minimal and not her usual style. 3/5 stars. ...more
May 04, 2009 Misfit rated it liked it · review of another edition
Aug 14, 2017 classic reverie rated it it was amazing · review of another edition
Shelves: favorites, daphne-du-maurier, 1900, english-writer, tragedy, own, romance, arthur-quiller-couch
I am a romantic but a truly hopeless one when it comes to fated loves from times past and especially of knights and their lady loves, which was Castle Dor but with an 1840 twist but finished 1961 The story is neither hard or easy read but it does need an attentive mind. I was amazed after reading the foreword, that Daphne was asked by Sir Arthur Quiller-Couch's daughter to finish his book. The story flowed like one author had written this, Daphne knew Sir Arthur from childhood and this might hav I am a romantic but a truly hopeless one when it comes to fated loves from times past and especially of knights and their lady loves, which was Castle Dor but with an 1840 twist but finished 1961 The story is neither hard or easy read but it does need an attentive mind. I was amazed after reading the foreword, that Daphne was asked by Sir Arthur Quiller-Couch's daughter to finish his book. The story flowed like one author had written this, Daphne knew Sir Arthur from childhood and this might have helped her achieve this feat. It is always interesting when an author's incomplete work is completed by another. Since Tristan and Iseult's story has different versions, until the last chapter I had no idea how it would end. Daphne's The House on the Strand had some kind of time travel to times long ago which was also a great read, as I read more and more of her novels I see her genius in storytelling. ...more
Jan 18, 2018 Rachel rated it it was ok
Having read a lot of Daphne du Maurier in the past, I was intrigued to find a novel I had not read before. A story of love and loss, with links to story parallels from the past. I found the read interesting, but felt it was over complicated at times. So making me feel unconnected to the main characters and not as interested in the plot. However, interesting to fans of Daphne du Maurier.
Oct 31, 2014 Judy rated it liked it · review of another edition
Shelves: books-from-1961, 20th-century-fiction
I don't recall how I first heard about Castle Dor. I think it was reviewed by one of my Goodreads friends. Since I am doing a completist reading of du Maurier's novels, I added it to my list.
Castle Dor was an incomplete novel by the very literary and august (according to my research) Sir Arthur Quiller-Couch. He died before finishing it. The du Mauriers were friends of Quiller-Couch so his daughter asked Daphne du Maurier to take over and write the rest of the book.
The story is based on the Celt
The story is based on the Celtic myth of Tristan and Iseult. Even I, who have only dabbled in mythology, know that tragic story of star-crossed lovers. In this version, which is set in the early 1840s near the Fowey River in the Cornish countryside, certain individuals unknowingly play the parts of the main characters in the legend.
The setting and the slipstream notion of people reliving a story from centuries earlier was the idea of Quiller-Couch. In the prologue he imparts the imaginations of a local doctor who spends the night waiting on a birth by standing on the ancient earthwork of a ruined Castle Dor and begins to fancy that he can perceive the sorrowful tale of those who lived there far in the past.
I loved the concept: "All England is a palimsest of such (quarrel, ancient feud, litigation), scored over with writ of hate and love, begettings of children beneath the hazels, appeals, curses, concealed travails." I was however challenged by the original author's rather florid and wordy style.
In several reviews readers have claimed that the continuation of the writing by du Maurier is seamless. I could tell right away when she took over, partly I suppose because I am familiar with her voice. Suddenly about a third of the way through I could read smoothly and easily without having to reread almost every sentence several times.
Then the book became a pageturner though it never lost that time travel essence. I ended up loving it and feeling as sad as if I hadn't know the lovers were doomed. I admired the skill with which she and Quiller-Couch placed the elements of the legend into the realities of life in the 1840s.
I am glad I read it. ...more
Jul 31, 2014 Peter rated it liked it
As a child I remember my maternal grandfather had a reasonably well stocked library and in it included most of the works of Arthur Quiller-Couch (Q). So it was with some interest that I discovered that this book had been started by Q and finished by du Maurier at the behest of Q's daughter so was intrigued as to how this collaboration would work.
Firstly let me say that it appears seamless and it is hard to see which author wrote what (good or bad depending on your taste) although there did seem As a child I remember my maternal grandfather had a reasonably well stocked library and in it included most of the works of Arthur Quiller-Couch (Q). So it was with some interest that I discovered that this book had been started by Q and finished by du Maurier at the behest of Q's daughter so was intrigued as to how this collaboration would work.
Firstly let me say that it appears seamless and it is hard to see which author wrote what (good or bad depending on your taste) although there did seem a noticeable quickening in the pace towards the end.
A chance meeting between Linnette LeWarne,a pretty but haughty young woman recently married to a much older man but still with dreams of romance, and a Bretton onion seller Amyot leads to an unlikely romance when Amyot rescues Linnette and her husband from a run-away coach accident with predictably disastrous results. This is interspersed with some good old Arthurian legends of a similar love affair between Tristan and Iseult.
The prose was generally excellent,the Cornish scenery was wonderfully portrayed particularly as the mist descends for the final curtain call as was the evocative easy going way of life therein. Although being Cornish myself may have some reflection on my opinion here. But that said at times it was fairly pedestrian almost scholarly in pace at times.The characters were generally only sketchily and there did seem to be an over reliance of an interest and knowledge of the Tristan Iseult affair all of which held back the overall feel of the novel IMHO.
On the whole I found this was an interesting collaboration if nor overly gripping one. ...more
Sep 16, 2009 Carrie Ridgeway rated it really liked it
A trajic love story centuries old repeated itself on the very lands where it previously occurred. Despite the efforts of Dr. Carfax to stop its progress - then came - just as with Tristan and Isuelt but for Amyst and Linnet.
May 12, 2011 Yuki rated it really liked it
An interesting re-telling of the story of Tristan and Iseult. The countryside of Cornwall is beautifully rendered and there are many allusions to the original story, so I found it to be fascinating. Cornwall is now on my bucket list for travel.
Jan 21, 2017 Sophie added it
Shelves: uk-author, didnt-finish, modern-classic
I love du Maurier, but I'm really not feeling this. The novel was originally started by Arthur Quiller-Couch and I can tell it's not DDM.
Feb 15, 2019 Sarah rated it it was ok
I finally got around to reading Castle Dor this month, after it sat on my bookshelf for about two years, as part of the UnreadShelfProject2019 on Instagram (February's book being a book that was received as a gift).
It was okay; in fact, very much a middle-ground read for me. I didn't hate it, but I didn't love it, either.
The thing about Castle Dor is that it was originally drafted by Sir Arthur Quiller-Couch, and du Maurier simply finished it after his death. It therefore doesn't read exactly as I finally got around to reading Castle Dor this month, after it sat on my bookshelf for about two years, as part of the UnreadShelfProject2019 on Instagram (February's book being a book that was received as a gift).
The thing about Castle Dor is that it was originally drafted by Sir Arthur Quiller-Couch, and du Maurier simply finished it after his death. It therefore doesn't read exactly as you would expect a du Maurier novel to read. Anyone familiar with du Maurier's work is likely to be able to pick out the passages written (or heavily edited) by her, and those of Quiller-Couch (as was the case for me).
The basic premise to Castle Dor is a re-telling of the Tristan & Iseult (Isolde) legend, set in 19th Century Cornwall. Tristan has become a Breton onion-seller by the name of Amyot Trestane, and Iseult the wife of a local publican named Linnet Lewarne. Their chance encounter sets in motion a series of events eerily similar to those known to legend.
A "modern" re-telling of the Tristan & Iseult legend is not a new idea, and Castle Dor certainly doesn't do a terrible job of it. But the novel as a whole felt underdone, and I found myself wishing for more depth and intrigue to the characters and events. I mean, I didn't really like any of the characters all that much: Amyot was too much of a mystery, Linnet too self-involved, Deborah's duplicity unconvincing, Mark Lewarne was only semi-present, and Dr Carfax was a bit of a prig.
I feel as though du Maurier tried to stay true to Quiller-Couch's original draft as much as possible, because that's very much how it reads. Had du Maurier completely re-written the story as her own I think there would have been a lot more depth to the story, and would therefore probably get a higher rating from me. ...more
Jul 22, 2018 Harriet rated it it was ok · review of another edition
This is not a good book. It is quite impenetrable at the start with one-dimensional, uninspiring characters and unnecessary, dull detail.
There is absolutely no explanation or background to how or why Linnet and Amyot fall in love - or indeed any real background to their characters at all. Perhaps this was a deliberate choice to render the novel similar to the original legend - but the consequence was that its hard to give a monkeys about what happens to them in the end.
I was hoping that there This is not a good book. It is quite impenetrable at the start with one-dimensional, uninspiring characters and unnecessary, dull detail.
I was hoping that there would be some good descriptions of the Cornish countryside to evoke the place. There weren't.
The two children (Mary and Johnny) and Dr Carfax were OK characters and an improvement on Amyot and Linnet. Linnet was annoying - I had no sympathy for her whatsoever.
The saving grace of the book (and the reason for my 2 stars) is that it picks up pace towards the end and I did keep reading to the end - so it can't have been all bad. There is also a dash of humour in places. My guess is that Du Maurier took over at around 65% - but that's a lot of book to wade through before it improves! ...more
Mar 31, 2019 Melanie Moore rated it it was amazing · review of another edition
Castle Dor was originally the unfinished work of British novelist Sir Arthur Auiller-Crouch, better known as "Q". His daughter sought out Daphne du Maurier to finish the tale. True to du Maurier's form, this novel was captivating. It's hard to tell where Q ended and du Maurier began. It is a modern retelling of the tragic love story of Tristan and Iseult. This novel is often overlooked, but if you a Daphne du Maurier fan, I highly recommend it!
Jul 13, 2017 Aimee Lauren rated it liked it · review of another edition
Whilst I think the beginning of the book was quite a different style to du Maurier (and I admit - it was rather difficult to get into!) I think du Maurier did an excellent job of picking up the plot and I thought the storyline, and the writing, improved about s third of the way in.
The ending even had a rather climatic du Maurier feel to it, so I think whilst it's not one of my favourites, I'm glad I've read it.
Jun 02, 2017 Karen Martin rated it liked it · review of another edition
I loved the detective work here, trying to decide where the original author stopped and Du Maurier took over. I realised I had reached this point when the reading became less of a slog and more of a page turner!
Some knowledge of Tristan and Iseult would be useful for the story, but not essential, as the old legend is woven in.
Interesting. Worth reading.
Review to follow soon on karenmartinreads.blogspot.co.uk
Jun 22, 2017 Sarah Thompson rated it liked it
Shelves: awlb-reading-challenge-2017
I stuck with this book as I have a lot of love for Daphne du Maurier's work. I have not read any of Sir Arthur Quiller-Couch's writing previously but Castle Dor seemed not to be written in Daphne's voice but in Sir Arthur's and I found it quite a dry and tedious read. Living in Cornwall I really enjoyed the geographical references and overall the story got better and became more readable towards the end but I'm glad it's over.
Jan 16, 2018 Linda Orvis rated it liked it · review of another edition
Shelves: historicals
Another of my pet authors. Castle Dor is a remake of the tale of Tristan and Isolde. It was published in 1961, and takes place in Cornwall, England (of course, as most of Daphne's books are). This is not my favorite du Maurier book, but it was worth reading, especially if you enjoy taking a trip to Cornwall along with her.
May 03, 2019 Mgb rated it it was ok · review of another edition
I have never been drawn to the story of Tristan and Iseult and this offshoot does not change that. I have enjoyed books by du Maurier but this one is low in the list. At the request of his daughter, she finishes the story started by the 'famous novelist' Q. Her part of the book is much more readable. The story is okay.
Apr 03, 2018 Rosemary Orme rated it it was ok · review of another edition
Shelves: 2018challenge
A retelling of Tristan and Isolda, which I studied for my Cornish Bard language exams. I really couldn't enter into the spirit of this, and fore knowledge of the tragic ending gave little incentive to struggle through it. Unusually for Daphne du Maurier, I did not enjoy this.
Sep 18, 2017 Lauren Hayward rated it it was ok · review of another edition
Shelves: gave-up
Got majorly fed up with this, you can tell du maurier had little to do with it as it doesn't reflect her style at all, got 1/4 of the way through but kept falling asleep
Aug 07, 2018 Kelly rated it really liked it · review of another edition
It wasn't quite as readable as the author's other works but still intriguing.
Sep 24, 2017 Hannah Stewart rated it it was ok · review of another edition
I understand that Daphne Du Maurier was only a co-writer on this book but it definitely didn't have her 'feel' to it.
May 31, 2017 Fang McGee rated it liked it · review of another edition
More miss than hit; the characterization was disappointingly wanting.
Apr 03, 2019 Jared Geraghty rated it liked it
Ugh I'm so conflicted on this one, because on the one hand it was really well written, but on the other I couldn't help but think it was so dull! Not much happened and the romance sprung from nowhere, and the plot felt rather rushed at the end though too drawn out at the beginning, and it was just weird to read. I loved the description, and the world building was beautiful, but the characters and plot were frankly a mess. But I did enjoy reading it somehow, which is why it gets a good rating. I Ugh I'm so conflicted on this one, because on the one hand it was really well written, but on the other I couldn't help but think it was so dull! Not much happened and the romance sprung from nowhere, and the plot felt rather rushed at the end though too drawn out at the beginning, and it was just weird to read. I loved the description, and the world building was beautiful, but the characters and plot were frankly a mess. But I did enjoy reading it somehow, which is why it gets a good rating. I know this doesn't really make sense, but I truly don't know how to review this one without contradicting myself! ...more
Mar 05, 2011 Bev rated it liked it · review of another edition
Shelves: fiction, gothic
Synopsis: "The castle and the hills around it had seen a doomed love affair before, but the impulsive young woman and the impressionable young lad from Brittany with whom she discovered a love heretofore denied her knew nothing of the past. They knew only the present--a present suddenly alive with enchantment, love and danger.
Linnet Lewarne at nineteen was married to a crotchety, wealthy man forty years her senior. One day there came to the seaport town where they lived a Breton, named Amyot Tre Synopsis: "The castle and the hills around it had seen a doomed love affair before, but the impulsive young woman and the impressionable young lad from Brittany with whom she discovered a love heretofore denied her knew nothing of the past. They knew only the present--a present suddenly alive with enchantment, love and danger.
Linnet Lewarne at nineteen was married to a crotchety, wealthy man forty years her senior. One day there came to the seaport town where they lived a Breton, named Amyot Trestane. Badly beaten by his captain, he decides to leave his ship and settle down in the village. Almost immediately he met and fell in love with Mrs. Lewarne, and she with him. Too young and too much in love, they continue to meet where and as they can...and suddenly tragedy comes, as it had to. Only the old doctor who brought Linnet Lewarne into the world recognizes--too late--the dark currents of the past which have brought the lovers together and will now tear them apart."
Castle Dor by Daphne du Maurier and Arthur Quiller-Couch is a very interesting synthesis on many levels. First of all, the story weaves the current story with the age-old love story of Tristan and Iseult. (I wish it hadn't been so long since I'd read that one...details were a bit fuzzy and would have been helpful.) There is a very real feeling that the past is intruding on the present and directing the events of the lives of the lovers as well as those around them. One is left wondering if it is meant as a sort of reincarnation or if it is another version of mystical time-travel (which du Maurier used in The House on the Strand). I lean towards the former.
There is also the blending of the work of Quiller-Couch and du Maurier. Quiller-Couch was very interested in Cornish history and the myths and legends of the area and he began this story. Upon his death, his daughter asked du Maurier, a friend of the family, to finish the story. The blending of the work is very well done and it is difficult to tell who is responsible for what. At a guess, I would say that du Maurier picked up the story at Book Two. I certainly do not believe that she is responsible for this image from the prologue: "The most ancient cirque of Castle Dor, deserted, bramble-grown, was the very nipple of a huge breast in pain, aching for discharge." Huh???
Overall, this is a very well told story of star-crossed lovers. The descriptions of the Cornish countryside is perfect (save for that bit above), especially in the latter chapters which I am sure is due to Ms. du Maurier. The working out of the legend/myth in the current love affair is done very adroitly and does not seem at all forced (as is sometimes the case when modern authors contrive a re-telling of an old story). And the minor characters are well-rounded and used to full effect. I do wish that Linnet Lewarne were a more sympathetic character. I don't remember details about the Tristan and Iseult story, but I do remember being more touched and involved with both of the lovers. In this version, I find myself sympathizing with Amyot far more than I do with with Linnet. A stronger sympathy for Linnet would bring my rating up to a full four stars, at the very least. As it is...three and half stars. It is very nearly outstanding on all counts.
This review was first posted on my blog My Reader's Block. Please request permission to repost any portion. Thanks. ...more
Apr 16, 2016 Ape rated it liked it
Shelves: history-1800s, england, history-fact-fiction
It's been years since I last read a Du Maurier novel. I do love her writing so perhaps it's an odd choice to go for Castle Dor as it's not really her novel. It's also my least favourite of all her work I've read to date. Really this is Sir Arthur Quiller-Couch's unfinished novel. He died before he could complete it and his daughter gave it to Du Maurier to complete. I think she's kept the voice consistent throughout, so she's done a good job. But it's not the Du Maurier that I love, and I did fi It's been years since I last read a Du Maurier novel. I do love her writing so perhaps it's an odd choice to go for Castle Dor as it's not really her novel. It's also my least favourite of all her work I've read to date. Really this is Sir Arthur Quiller-Couch's unfinished novel. He died before he could complete it and his daughter gave it to Du Maurier to complete. I think she's kept the voice consistent throughout, so she's done a good job. But it's not the Du Maurier that I love, and I did find it a bit archaic, slow and dry. I'm glad I read it once, and certainly for any fan of Du Maurier, it's got to be read, but I definately wouldn't suggest anyone new to her books starts here.
Set in the 1800s in Cornwall, it's about Linnet Lewarne, who is about 19 and recently married to a 60 year old inn landlord. So immediately we all realise there's going to be problems there. In comes Breton sailor-onion seller Amyot, and it's essentially love at first sight. Although there may be greater forces at play, as this is a retelling of the Tristan and Iseult story. But it's not just a retelling because all the way through we're being told about this connection and how places in the surrounding landscape prove that it happened once and it happened there. There is this horrible character (although I think he's meant to be likeable) called Dr Carfax who is keen on place name history and classics and is researching the history's area. He's also long convinced that those two are under the influence of Tristan and Iseult. Why did I think he was horrible? He takes it upon himself to interfere in people's lives. Really interfere, to stop them doing things that he thinks is wrong, when bottom line is: they're adults and it's none of his business. And in the end what does his interfering gain? Well, he stops Linnet and Amyot running off together but he also sees them both dead - Amyot from an infection from the doctor's pocket knife, and Linnet from a drugged drink her husband gave her - hoping it would make her sleep, but actually no one knew what was in the bottle in the first place! Oh, it may be all rose tinted on looking back at the good old rural days, with pretty scenary and simple folk, but I certainly couldn't live in such a claustrophobic environment where not only does everyone know your business but they'll be meddling in it as well.
I'll finish with a rather pompous quote from Dr Carfax who is showing a new French friend, a notary, around the area earlier on in the book. It's so random that it just made me laugh.
"To the left, thick hazel overhung them. 'A good year for nuts,' said Doctor Carfax, indicating some clusters with his whip. 'There's a saying in these parts that a good year for nuts is a good year for bastards...'" (p.66) And on that random little gem, I'm rather pleased I've finished this book now. ...more
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Historical > Historical Fiction
European Literature > British Literature
About Daphne du Maurier
Daphne du Maurier
If Daphne du Maurier had written only Rebecca, she would still be one of the great shapers of popular culture and the modern imagination. Few writers have created more magical and mysterious places than Jamaica Inn and Manderley, buildings invested with a rich character that gives them a memorable life of their own.
In many ways the life of Daphne du Maurier resembles a fairy tale. Born into a fami If Daphne du Maurier had written only Rebecca, she would still be one of the great shapers of popular culture and the modern imagination. Few writers have created more magical and mysterious places than Jamaica Inn and Manderley, buildings invested with a rich character that gives them a memorable life of their own.
In many ways the life of Daphne du Maurier resembles a fairy tale. Born into a family with a rich artistic and historical background, the daughter of a famous actor-manager, she was indulged as a child and grew up enjoying enormous freedom from financial and parental restraint. She spent her youth sailing boats, travelling on the Continent with friends, and writing stories. A prestigious publishing house accepted her first novel when she was in her early twenties, and its publication brought her not only fame but the attentions of a handsome soldier, Major (later Lieutenant-General Sir) Frederick Browning, whom she married.
Her subsequent novels became bestsellers, earning her enormous wealth and fame. While Alfred Hitchcock's film based upon her novel proceeded to make her one of the best-known authors in the world, she enjoyed the life of a fairy princess in a mansion in Cornwall called Menabilly, which served as the model for Manderley in Rebecca.
Daphne du Maurier was obsessed with the past. She intensively researched the lives of Francis and Anthony Bacon, the history of Cornwall, the Regency period, and nineteenth-century France and England. Above all, however, she was obsessed with her own family history, which she chronicled in Gerald: A Portrait, a biography of her father; The du Mauriers, a study of her family which focused on her grandfather, George du Maurier, the novelist and illustrator for Punch; The Glassblowers, a novel based upon the lives of her du Maurier ancestors; and Growing Pains, an autobiography that ignores nearly 50 years of her life in favour of the joyful and more romantic period of her youth. Daphne du Maurier can best be understood in terms of her remarkable and paradoxical family, the ghosts which haunted her life and fiction.
While contemporary writers were dealing critically with such subjects as the war, alienation, religion, poverty, Marxism, psychology and art, and experimenting with new techniques such as the stream of consciousness, du Maurier produced 'old-fashioned' novels with straightforward narratives that appealed to a popular audience's love of fantasy, adventure, sexuality and mystery. At an early age, she recognised that her readership was comprised principally of women, and she cultivated their loyal following through several decades by embodying their desires and dreams in her novels and short stories.
In some of her novels, however, she went beyond the technique of the formulaic romance to achieve a powerful psychological realism reflecting her intense feelings about her father, and to a lesser degree, her mother. This vision, which underlies Julius, Rebecca and The Parasites, is that of an author overwhelmed by the memory of her father's commanding presence. In Julius and The Parasites, for example, she introduces the image of a domineering but deadly father and the daring subject of incest.
In Rebecca, on the other hand, du Maurier fuses psychological realism with a sophisticated version of the Cinderella story. The nameless heroine has been saved from a life of drudgery by marrying a handsome, wealthy aristocrat, but unlike the Prince in Cinderella, Maxim de Winter is old enough to be the narrator's father. The narrator thus must do battle with The Other Woman—the dead Rebecca and her witch-like surrogate, Mrs Danvers—to win the love of her husband and father-figure. ...more
Books by Daphne du Maurier
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Quotes from Castle Dor
"She's dearer than life itself, that's all I know." — 4 likes
"That this exceptionally scholarly man whose judgments, always rich and sensitive, though sometimes austere, should have embarked on an intensely romantic retelling of the old Cornish legend of that famous pair of tragic lovers, Tristan and Queen Iseult, is intriguing in itself. But what makes it even more fascinating is that Daphne du Maurier, asked by "Q" 's daughter long after her father's death to finish this novel that he had set aside "near the end of a chapter, halfway through," did so in such a skillful fashion that it is impossible to guess with any certainty the exact point at which she began to write. She says, in a modest foreword, that she "could not imitate 'Q''s style… that would have been robbing the dead," but she had known him when she was a child, remembered him as a genial host at many a Sunday supper, and "by thinking back to conversations long forgotten" she could recapture something of the man himself and trust herself to "fall into his mood." — 0 likes


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HomeOpinionGeneral Buratai: Why Leadership Is Key In The War Against Insurgency...
General Buratai: Why Leadership Is Key In The War Against Insurgency , By David Onmeje
Leadership is an art meant for those that understand the intricacies involved in bringing about positive change in every society. I say this for the fact that in Nigeria, one of the banes of our numerous challenges remains the lack of sound and credible leadership by those saddled with the responsibility of translating our hopes and aspirations into tangible realities.
The recent commendation of the Air Task Force, Operation Lafiya Dole by the Chief of Army Staff, Lt Gen TY Buratai for playing a decisive role in repelling the Boko Haram attack in Marte, Borno State, recently brings to fore why leaders in Nigeria must rise to the occasion when the need arises.
The action of Lt. Gen. Tukur Buratai is such that should inspire hope in Nigeria on the commitment of the security agencies towards addressing the security threats in the country. I can't but commend this display of professionalism in the prosecution of the war against Boko Haram insurgency in Nigeria. The synergy level that has existed amongst the Armed Forces is exceptional and contrary to the insinuations in certain quarters of a lack of cooperation and other acts that undermines the winning of the war against insurgency in the country.
I know for sure that the ground troop in warfare requires air support in advancing because of its considerable advantage. The Army increasingly views airpower as indispensable to its warfighting concepts. It seeks mechanisms to ensure that it is available and responsive to the needs of the land forces. This much has been exhibited in the war against insurgency in Nigeria. Needless, I mention that the recent commendation by the Chief of Army Staff indeed buttresses this strategic alliance which is an indication that the overall operational strategy deployed is yielding the desired results.
For those of us conversant with the dynamics involved in asymmetric warfare, I say that the Chief of Army Staff's posture has indeed indicated that the Boko Haram war would come to an end sooner than we think for all the indices shows so.
I am of the firm conviction that the appointment of Lt. Gen. Tukur Buratai as Chief of Army Staff was not an accident as he has continually provided that leadership that turned the tides against the Boko Haram insurgents. And the fact that there has been no grandstanding or taking personal credit is an attribute of a leader that understands that success is a function of teamwork and not individual efforts.
This also brings to the fore that a true leader leads even within leadership. Lt. Gen. Tukur Buratai has been providing leadership even amongst equals which also confirms the Nigerian Army's contributory role to the war against insurgency. This is noble and must be emulated by others in the security sector in the prosecution of the war against insurgency in Nigeria.
What I am saying, in essence, is that credit must be given to whom it due. For example, the Inspector General of Police must commend the Nigerian Army whenever successes are recorded in internal security operations. And likewise, the Chief of Air Staff must also commend the Nigerian Army whenever successes are recorded in joint operations in the prosecution of the war against insurgency and other acts of criminalities in the country.
The Nigerian Air force case is pronounced because they must work with the Nigerian Army to address the security threats in the country. Both institutions must sustain this alliance which remains crucial to ending the Boko Haram insurgency in Nigeria. Make no mistakes; I am not by any stretch of imagination insinuating that there hasn't been that synergy between the Nigerian Army and the Nigerian Air Force. However, instead, I am bringing out the positive gains that would be attained if the present synergy is sustained and built.
This piece is consequently an appeal to the Armed Forces' leadership in Nigeria to see the task at hand as a collective endeavour rather than a solo run. There must be a synergy that leaves no room for individual glory. That has remained the norm in successful outings in warfare globally. The Air Force uses air power to clear the path for ground troops and provide that critical cover in battle.
This example provided by the Chief of Army Staff is indeed noteworthy. It must awaken us to the reality that for Nigeria to work, we must imbibe the principle of structural functionalism which requires all playing their role in the task at hand. We must also realize that the Nigerian Army cannot do it all alone. Like the Nigerian Airforce played a decisive role in repelling the Boko Haram attack in Marte to support the ground troops.
We must learn to commend the security agencies when they do well to encourage them to carry out more exploits in addressing the security challenges in the country. I can bet that if this is the order of the day, Nigeria can defeat acts of terrorism in no distant time.
At this point, my commendation goes to the leadership of the country's security agencies for their great sacrifices thus far. It can only charge them to continue to give in their best in service to our fatherland.
I also want to commend the Chief of Army Staff, Lt. Gen. Tukur Buratai who has exhibited credible leadership since he assumed office. His level of professionalism and dedication to duty has seen the numerous successes recorded in the prosecution of the war against insurgency and other criminalities acts across the country.
The leadership qualities of the Lt. Gen. Tukur Buratai were also in focus on the involvement of the Nigerian Army in internal security operations across the country in support of the efforts of the Nigerian Police Force. This is also commendable and gives us a peep that leadership entails that what you do has a far more significant impact than what you say. This is the Tukur Buratai style that should be imbibed by leaders across Nigeria in our quest for the Nigeria of our dreams.
As I mentioned earlier, the commendation by the Chief of Army Staff might seem a simple task. However, it goes beyond that for it is indeed strategic and sends a strong message to those in the business of perpetuating crime and criminalities in the country that there is a synergy amongst the country's security agencies. And no matter how hard they try; the ranks of the security agencies can't be broken.
I can go on non-stop. But the take away is that with sound leadership, Nigeria would defeat its adversaries. The Chief of Army Staff has indeed done well so far, and I expect that other security heads would toe this path in the country's overall interest.
The war against insurgency has recorded tremendous success so far, and I am of the firm conviction that the end must be seen in our time with the efforts so far. It consequently behoves on us to see to the fact that the efforts of our security agencies are appreciated. We should do well not to join the bandwagon of mischievous elements whose actions are propelled by the support they have been receiving from crisis merchants in their despicable and desperate plot to cause Nigeria's disintegration.
Onmeje is the Managing Director, TheNigerian News and is based in London.
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The Discoverer
More MS Students Participate in Copa Columbus 2019
Jeronimo
Colegio Aleman, and Colegio San Ignacio participating in a soccer game in the 2019 middle school Copa Columbus.
Jeronimo Restrepo Toro, Discoverer Staff Writer
New changes were implemented to Middle School's Copa Columbus this year, held on October 23-26, to provide new opportunities for each athlete to succeed in its respective sport.
For some of the sports, the school had 2 teams, one for 6th grade and one for 7th and 8th grade. In the past years, the school had tried to implement or make this new feature a thing. This year it was official.
"It impacted the students, mostly in 6th grade, it gave them an opportunity to play, because in the last years if you were in 6th grade you did not have an opportunity to play or participate, and that will impact a lot in the future years, I think it will help improve the students mindset about Copa, and about sports in general." Freddy Badillo, TCS Athletic Director, said.
The school's team had improving results compared to past years. The 2 categories of boys and girls soccer, were in the top 3, the 6th-grade girls won. Basketball, boys, and girls also got into the top 3. Volleyball teams also reached top positions. The 6th graders had good results this Copa, they felt inspired and this will affect the next generation of sports in Middle School.
"I think I felt very happy because we played in the elementary soccer field, all elementary was supporting us, it was really cool because you felt important, you felt the pressure in the game so it helped us a lot to get to third place, and also it helped the rest of the 6th graders in my team", Luciano Restrepo, 6th grade boys soccer captain, said.
The students felt good and happy during the games, and also learned to adapt to the new changes.
"For the most part, the students had a smile on their faces. Not only the teams were happy, but the entire school enjoyed this Copa, the results don't matter, what matters is participation and sportsmanship, this is a great year, and the students learned a lot from this past games."
Every year, more and more people are participating in Copa Columbus. And the expectations for the teams increases also.
"Saying this, with all the students that participated this year in Copa, I think we exceeded the expectations for this year Copa," Badillo said.
COSMUN 2019 News: Day 2
COSMUN News 2019: Day 1
A Tribute to David Watson
Lightning Blue Lizards Robotics Team Season Recap
You Grow Girl
Face to Face Classes Will Reopen at TCS
Homecoming: Students Safe Return to Campus
Technology's Effect on Students Health during COVID-19 Pandemic
Directives Implement New Distance Learning Schedule
8th Grade STUCCO Elections
Middle School, MiniGIN Group Collaborates with Llenando Botellas de Amor
Thanksgiving Assembly Rescheduled Because of Nationwide Protests
New Library Platform Offers 1000's of Book Titles
"Eggstronaut": More Than Just a Project
The Columbus School News Portal


text:
\section{Introduction}
A continuous map $f:X\rightarrow Y$ is called proper if $f^{-1}(C)$ is compact for all compact subsets $C$ of $Y$. A homotopy of proper maps is a homotopy $F:[0,1]\times X\rightarrow Y$ such that $F$ is a proper map. The assumption that a homotopy is a homotopy of proper maps is stronger than the assumption that the homotopy is homotopy through proper maps, i.e. the assumption that the maps $F_t:X\rightarrow Y$ are proper for every $t\in[0,1]$. A simple example of a homotopy through proper maps that is not a homotopy of proper maps is the map $F:[0,1]\times \mR\rightarrow \mR$ defined by $F(t,x)=(1-t)x^2+x$. To see this note that the sequence $(1-\frac{1}{n},-n)$ is unbounded, but $F(1-\frac{1}{n},-n)=0$. This example is closely related to the compactness issues discussed in~\cite{Geiges:2018fo}.

We denote by $[X,Y]$ the set of (unbased) homotopy classes of maps from $X$ to $Y$ and with $[X,Y]_{\mathrm{prop}}$ the set of (unbased) homotopy classes of proper maps. For the set of homotopy classes of based maps between pointed spaces we write $\langle X,Y\rangle$.

In~\cite{Abbondandolo:2018fb} we classified the homotopy classes of proper Fredholm maps of Hilbert manifolds into its model (real and separable) Hilbert space in terms of a suitable notion of framed cobordism. This classification uses an infinite-dimensional and proper analogue of the Pontryagin-Thom collapse map, which is due to Elworthy and Tromba~\cite{Elworthy:1970vp}, see also the paper of G\k{e}ba~\cite{Geba:1969df}. The existence of the collapse map hinges on the fact that an infinite dimensional Hilbert space is diffeomorphic to the Hilbert space minus a point. This is of course not true for a finite dimensional vector space and the construction does not work in this setting. As we will discuss in Section~\ref{framed}, even though the framed cobordism class of a regular value is an invariant of the homotopy class of a proper map in the finite dimensional setting, the framed cobordism class is not able to distinguish all proper homotopy classes of proper maps into $\mathbb{R}^k$, nor do all framed submanifolds come from proper maps. Thus there does not exist a finite dimensional proper Pontryagin-Thom construction, which is why we are not able to compute $[E,\mR^k]_{\mathrm{prop}}$ for all open finite dimensional manifolds $E$ using a Pontryagin-Thom collapse map. In this paper we are content with the classification of $[E,\mR^k]_{\mathrm{prop}}$ where $E$ is a real vector bundle over a compact Hausdorff space $M$. This classification does not use a Pontryagin-Thom collapse map.

\begin{theorem}
\label{maintheorem}
Let $E\rightarrow M$ be a normed vector bundle over a compact Hausdorff space $M$. Denote by $S(E)\rightarrow M$ the associated sphere bundle of unit vectors. Then the set $[E,\mathbb{R}^k]_{\mathrm{prop}}$ is in bijective correspondence with $[S(E),S^{k-1}]$.
\end{theorem}

We have the following corollary of Theorem~\ref{maintheorem} by taking $M$ to be a point and using the fact that based and unbased homotopy classes of maps from spaces to positive dimensional spheres coincide, cf.~\cite[Section 4A]{Hatcher_Topology}.

\begin{corollary}
\label{cor:sphere}
$[\mR^n,\mR^k]_{\mathrm{prop}}$ is in bijective correspondence with $[S^{n-1},S^{k-1}]$. Thus if $n>1$ and $k>1$ we have that $[\mR^n,\mR^k]_{\mathrm{prop}}$ is in bijection with $\pi_{n-1}(S^{k-1})$. The set $ [\mR^n,\mR]_{\mathrm{prop}}$ has two elements if $n>1$ and four elements if $n=1$.
\end{corollary}

A proper map between non-compact and locally compact Hausdorff spaces extends to a continuous map between the one point compactifications by sending infinity to infinity. Similarly a homotopy of proper maps induces a homotopy in the one-point compactification.

The one point compactification of a real vector bundle $E\rightarrow M$ over a compact Hausdorff space $M$ equals the Thom space $\Th(E)$ of the vector bundle and the one point compactification of $\mR^k$ is homeomorphic to $S^k$ by stereographic projection. Thus we obtain a map $Q:[E,\mR^k]_{\mathrm{prop}}\rightarrow \langle\Th(E),S^k\rangle$. In Section~\ref{sec:suspensions} we show that the map $Q$ is bijective in a range. If $E=\mR^n$ the map $Q$ is nothing but the suspension $\pi_{n-1}(S^{k-1})\rightarrow \pi_n(S^k)$ under the identification of $[\mR^n,\mR^k]_{\mathrm{prop}}$ and $\pi_{n-1}(S^{k-1})$ of Corollary~\ref{cor:sphere}.

For $l$ sufficiently the sets $[E\oplus \mR^l,\mR^{k+l}]_{\mathrm{prop}}$ and $[E\oplus \mR^{l+1},\mR^{k+l+1}]$ are in bijection. Thus it makes sense to define the stable proper homotopy classes as $[E,\mR^k]^S_{\mathrm{prop}}=\lim_{l\rightarrow \infty}[E\oplus \mR^l,\mR^{k+l}]$, which are in bijection with the stable cohomotopy groups $\pi^k_S(\Th(E))$, cf.~Corollary~\ref{cor:stable}. Using Atiyah duality we obtain the following result.

\begin{theorem}
\label{thm:atiyah}
Let $M$ be an $m$-dimensional compact manifold with boundary $\partial M$. Let $E$ be the normal bundle of $M$ of some embedding of $M$ into $\mR^{m+n}$. Then there is a bijection of $[E,\mR^k]^S_{\mathrm{prop}}$ with $\pi_{n+m-k}^S(M/\partial M)$.
\end{theorem}

In Section~\ref{sec:speculation} we speculate on the classification problem in the case that $E$ is an arbitrary open manifold.

\subsection{Acknowledgements} I would like to thank Alberto Abbondandolo, Hansj\"org Geiges, Gijs Heuts, and Federica Pasquotto for discussions on the content of this paper. This research was supported by NWA startimpuls - 400.17.608.
\section{The proof of Theorem~\ref{maintheorem}}

For the remainder of the paper $E\rightarrow M$ denotes a normed real vector bundle over a compact Hausdorff space $M$. The associated sphere and disc bundles of radius $R>0$ are
$$S_R(E)=\{v\in E\,|\,\norm{v}=R\}\, \quad\text{and}\quad B_R(E)=\{v\in E\,|\,\norm{v}<R\}.$$
We write $S(E)$ for $S_1(E)$ and $B(E)$ for $B_1(E)$.

Given a homotopy $F:[0,1]\times S(E)\rightarrow S^{k-1}$ we define the homotopy $PF:[0,1]\times E\rightarrow \mR^k$ of proper maps by
$$
PF(t,v)=\begin{cases}
\norm{v}F\left(t,\frac{v}{\norm{v}}\right)\qquad &v\not=0\\
0\qquad &v=0.
\end{cases}
$$
Compact subsets of $E$ are characterized as follows: A subset $K\subseteq E$ is compact if and only if it is closed and bounded. Here bounded means that $K\subseteq B_R(E)$ for some $R>0$. Let us prove that $PF$ is proper. Let $C\subseteq \mR^k$ be compact. Then $C$ and hence $PF^{-1}(C)$ are closed as $PF$ is continuous. Compact subsets of $\mR^k$ are the closed and bounded subsets, hence the set $C$ is contained in $B_r(\mR^k)$ for some $r>0$. Thus $PF^{-1}(C)$ is a closed subset contained the bounded set $[0,1]\times B_r(E)$ hence is compact. We conclude that that $PF$ is proper.

The same construction assigns to a map $f:S(E)\rightarrow S^{k-1}$ a proper map $Pf:E\rightarrow \mR^k$ and  it therefore induces a map $P:[S(E),S^{k-1}]\rightarrow [E,\mR^k]_{\mathrm{prop}}$.

We will show that $P$ is bijective. Let us start with the injectivity. We need to show that if $g_0=Pf_0$ and $g_1=Pf_1$ are homotopic as proper maps, that this implies that $f_0$ and $f_1$ are homotopic. Let $G:[0,1]\times E\rightarrow \mR^k$ be a homotopy  of proper maps between $g_0$ and $g_1$. Then for any $r>0$ there exists an $R>0$ such that $G^{-1}(B_r(\mR^k))\subseteq [0,1]\times B_R(E)$. It follows that for any $v\in E$ with $\norm{v}=R$ that $G(t,v)\not=0$. The map $F:[0,1]\times S(E)\rightarrow S^{k-1}$ given by
$$
F(t,x)=\frac{G(t,Rx)}{\norm{G(t,Rx)}},
$$
is a homotopy between $f_0$ and $f_1$ hence $P$ is injective.

To show that $P$ is surjective, we need to show that, given a proper map $g:E\rightarrow \mR^k$, there exists a homotopy of proper maps from $g$ to $Pf$, where $f$ is some map $f:S(E)\rightarrow S^{k-1}$. As $g$ is proper, there exists an $R>0$ such that $g^{-1}(B(\mR^k))\subseteq B_R(E)$. The sphere bundle $S_R(E)$ is compact, hence there exists an $r\geq 1$ such that
$$
1\leq\norm{g(v)}\leq r,\quad\text{for all}\quad v\in S_R(E).
$$
Consider the map $h:\mR^k\rightarrow \mR^k$ defined by
$$
h(x)=\begin{cases}x\qquad &\norm{x}\leq 1\\
\frac{x}{\norm{x}}\qquad &1\leq \norm{x}\leq r\\
\frac{x}{r}\qquad &r\leq \norm{x},
\end{cases}
$$
and define $g_1:E\rightarrow \mR^k$ by $g_1(v)=h\circ g(Rv)$. As $h$ is homotopic as proper map to the identity via $(t,x)\mapsto (1-t) h(x)+x$ it follows that he map $g_1$ is proper homotopic to $g$. Note that $g_1(B(E))\subseteq B(\mR^k)$ and $g_1(S(E))\subseteq S^{k-1}$. Define $G_1:[0,1]\times E\rightarrow E$ by
$$
G_1(t,v)=\begin{cases}\frac{\norm{g_1(v)}}{\norm{g_1\left((1-t) v+\frac{t}{\norm{v}}v\right)}}g_1\left((1-t)v+\frac{t}{\norm{v}}v\right) \qquad &\norm v\geq 1\\
g_1(v)\qquad &\norm v\leq 1.
\end{cases}
$$
The equation $\norm{G_1(t,v)}=\norm{g_1(v)}$ implies that $G_1^{-1}(B_s(\mR^k))\subseteq [0,1]\times g_1^{-1}(B_s(\mR^k))$ for all $s$ and hence that $G_1$ is proper if $g_1$ is. Thus $g_2:E\rightarrow \mR^k$ given by $g_2(x)=G_1(1,x)$ is a proper map that is proper homotopic to $g$. Let $f:S(E)\rightarrow S^{k-1}$ be the map obtained by restriction of $g_2$. Consider
$$
G_2(t,v)=(1-t)g_2(v)+t Pf(v).
$$
We want to prove that $G_2$ is proper. Note that $G_2$ sends $\overline{ [0,1]\times B(E)}$ to $\overline{B(\mR^k)}$ and $[0,1]\times (E\setminus B(E))$ to $\mR^k\setminus B(\mR^k)$. The map $G_2\bigr|_{\overline{[0,1]\times B(E)}}: \overline{[0,1]\times B(E)}\rightarrow \overline{B(\mR^k)}$ is proper as the domain is compact. We conclude that $G_2$ is proper if and only if $$G_2\bigr|_{[0,1]\times (E\setminus B(E))}: [0,1]\times (E\setminus B(E))\rightarrow \mR^k\setminus B(\mR^k)$$ is proper.

For this it is sufficient to show that for all $s>1$ there exists an $S>1$ such that $G_2^{-1}(B_s(\mR^k)\setminus B(\mR^k))\subseteq [0,1]\times (B_S(E)\setminus B(E))$. Note that for all $(t,v)\in [0,1]\times (E\setminus B(E))$ we have that
$$
G_2(t,v)=\left((1-t)\norm{g_1(v)}+t \norm v\right) f\left(\frac{v}{\norm v}\right).
$$
Consider all $(t,v)\in [0,1]\times (E\setminus B(E))$ such that $\norm{G_2(t,v)}\leq s.$ As $\norm {f(\frac{v}{\norm v})}=1$ this amounts to
$$
(1-t)\norm {g_1(v)}+t\norm{v}\leq s.
$$
Suppose the set of solutions of this equation is not contained in $[0,1]\times (B_S(E)\setminus B(E))$ for any $S$. Then we have a sequence $(t_n,v_n)$ of solutions such that $\norm{v_n}\geq n$. Without loss of generality we take a subsequence such that $t_n$ converges to $t$ by the compactness of $[0,1]$. This subsequence will also satisfy $\norm{v_n}\geq n$. If $t>0$, then there exists an $N$ such that for all $n\geq N$ we have $t_n>\frac{t}{2}$ and
$$
\norm{v_n}\leq \frac{1}{t_n}\left((1-t_n)\norm {g_1(v_n)}+t_n\norm{v_n}\right)\leq \frac{2s}{t},
$$
which contradicts the unboundedness of $v_n$. If $t=0$ then there exists an $N$ such that for all $n\geq N $ the sequence satisfies $t_n<\frac{1}{2}$ and
$$
\norm {g_1(v_n)}\leq\frac{1}{(1-t_n)}\left((1-t_n)\norm {g_1(v_n)}+t_n\norm{v_n}\right)\leq 2s.
$$
The sequence $g_1(v_n)$ is therefore bounded and as the map $g_1$ is proper it follows that the sequence $v_n$ is also bounded. This contradicts the assumption that $v_n$ is unbounded. This means that $G_2\bigr|_{[0,1]\times (E\setminus B(E))}$ is proper. Thus $Pf$ is proper homotopic to $g$ and $P:[S(E),S^{k-1}]\rightarrow [E,\mR^k]_{\mathrm{prop}}$ is surjective. We have already shown that $P$ is injective and Theorem~\ref{maintheorem} follows.

\section{The one point compactification and stable (co)homotopy}
\label{sec:suspensions}

Recall that the one point compactification of a non-compact, locally compact Hausdorff space $X$ is the space $X^*=X\cup\{\infty\}$ equipped with the following topology. All open sets $U$ of $X$ are declared open in $X^*$ along with all sets of the form $(X\setminus C)\cup \{\infty\}$ for all compact sets $C$ in $X$. Proper maps between non-compact, locally compact Hausdorff spaces induce continuous maps between the one point compactifications by imposing that $\infty$ is mapped to $\infty$. A homotopy of proper maps $F:[0,1]\times X\rightarrow Y$ induces a continuous map $F^*:([0,1]\times X)^*\rightarrow Y^*$. But $([0,1]\times X)^*\cong [0,1]\times X^*/[0,1]\times \{\infty\}$. By the universal property of the quotient topology we therefore also obtain a continuous map $F^*:[0,1]\times X^*\rightarrow Y^*$ which sends every $(t,\infty)$ to $\infty$. A homotopy of proper maps between unbased spaces is mapped to a based homotopy between the based spaces. The one point compactification of a vector bundle $E\rightarrow M$ over a compact Hausdorff space $M$ is the Thom space of the bundle and we will write $\Th(E):=E^*$. Stereographic projection shows that $(\mR^k)^*\cong S^k$ and in more generality it holds that $\Th(E)\cong \overline{B(E)}/S(E)$. As was mentioned in the introduction, the map that forgets the basepoint induces a bijection between $\langle \Th(E),S^k\rangle$ and $[\Th(E),S^k]$ if $k\geq 1$. Thus from the one point compactification we obtain a map $Q:[E,\mR^k]_{\mathrm {prop}}\rightarrow [\Th(E),S^k]$.

In our setting there are three suspension maps, which we all denote by $S$. To a map $g:S(E)\rightarrow S^{k-1}$ we associate the map $S f:S(E\oplus \mR)\rightarrow S^k$ by
$$
Sg(v,s)=\begin{cases}
(\norm v g\left(\frac{v}{\norm v}\right),s)\qquad &v\not=0\\
(0,s)&v=0.
\end{cases}
$$
To a map $f:\Th(E)\rightarrow S^k$ we associate the map $Sf:\Th(E\oplus \mR)\rightarrow S^{k+1}$  via the same formula and to a proper map $f:E\rightarrow \mR^k$ we associate the proper map $Sf:E\oplus \mR\rightarrow \mR^{k+1}$, by $Sf(x,s)=(f(x),s)$. The following diagram is commutative
\begin{equation}
\label{diag}
\begin{gathered}
\xymatrix{
[S(E),S^{k-1}]\ar[d]_S\ar[r]^P&[E,\mR^k]_{\mathrm{prop}}\ar[d]_S\ar[r]^Q&[\Th(E),S^k]\ar[d]_S\\
[S(E\oplus\mR),S^{k}]\ar[r]^P&[E\oplus\mR,\mR^{k+1}]_{\mathrm{prop}}\ar[r]^Q&[\Th(E\oplus\mR),S^{k+1}].
}
\end{gathered}
\end{equation}
In the proof of Theorem~\ref{maintheorem} we saw that the maps $P$ are bijections. We wonder when the other maps in the diagram are bijective.

Let us now assume that $k\geq 2$, that $M$ is a finite connected CW-complex of dimension $m$, and that $E$ is a vector bundle of rank $n$. Since $k\geq 2$ based and unbased homotopy classes into $S^{k-1}$ and $S^k$ coincide, as well as based or unbased proper homotopy classes into $\mR^k$ and $\mR^{k+1}$. We denote by $$\pi^{k-1}(S(E)):=\langle S(E),S^{k-1}\rangle$$ the $(k-1)$-th cohomotopy set of $S(E)$. We refer to \cite[Chapter VII]{Hu:1959ve} for information on the cohomotopy sets we use below. The cohomotopy set $\pi^{k-1}(S(E))$ is not always a group, but only if $m+n\leq 2k-3$. We investigate the long exact sequence of the pair $(\overline{B(E)},S(E))$ if $m+n\leq 2k-3$.
\begin{equation}
\label{eq:longexact}
\pi^{k-1}(\overline{B(E)})\rightarrow \pi^{k-1}(S(E))\xrightarrow{\delta} \pi^k(\overline{B(E)},S(E))\rightarrow \pi^k(\overline{B(E)})
\end{equation}
Since $\overline{B(E)}$ deformation retracts to $M$ and $S^{k-1}$ is $(k-2)$-connected, we see that if $m\leq k-2$ there are  isomorphisms
$$
\pi^{k-1}(\overline{B(E)})\cong\pi^{k-1}(M)\cong 0\quad\text{ and}\quad \pi^{k}(\overline{B(E)})\cong\pi^{k}(M)\cong 0.
$$
Thus we conclude that for $2k\geq m+3+\max(n,m+1)$ there is an isomorphism $\pi^{k-1}(S(E))\cong\pi^k(\overline{B(E)},S(E))$. The relative cohomotopy set is the cohomotopy set of the quotient for nice spaces, thus $\pi^{k}(\overline{B(E)},S(E))=\pi^k(\overline{B(E)}/S(E))=\pi^k(\Th(E))$. The coboundary map is an isomorphism $\pi^{k-1}(S(E))\cong\pi^k(\Th(E))$ in the dimension range. Let us consider the based version of Diagram~\eqref{diag}
\begin{equation}
\begin{gathered}
\xymatrix{
\pi^{k-1}(S(E)) \ar[d]_S\ar[r]^{QP}&\pi^{k}(\Th(E))\ar[d]_S\\
\pi^{k}(S(E\oplus \mR))\ar[r]^{QP}&\pi^{k+1}(\Th(E\oplus R)).
}
\end{gathered}
\end{equation}
The horizontal maps can be identified with the coboundary map $\delta$ in~\eqref{eq:longexact} and therefore the horizontal maps are isomorphisms in the right dimension range. Freudenthal's suspension Theorem, cf.~\cite{Kochman:1996ir}, states that if $m+n\leq 2k-2$, that the suspension map $\pi^k(\Th(E))\rightarrow \pi^{k+1}(S\Th(E))\cong\pi^{k+1}(\Th(E\oplus\mR))$ is an isomorphism. Combining all information we have now gives us
\begin{theorem}
\label{thm:iso}
Let $M$ be a finite $CW$ complex of dimension $m$ and $E$ a vector bundle over $M$ of rank $n$. Let $k\geq 2$ and suppose that $2k\geq m+3+\max(n,m+1)$. Then all maps in Diagram~\eqref{diag} are bijections.
\end{theorem}

This theorem expresses a stability phenomenon: For all $l$ sufficiently large the map $Q$ induces bijections $[E\oplus \mR^l,\mR^{k+l}]_{\mathrm{prop}}\rightarrow [E\oplus \mR^{l+1},\mR^{k+l+1}]_{\mathrm{prop}}$. We define the stable homotopy classes of proper maps as
$$
[E,\mathrm{\mR^k}]^S_{\mathrm{prop}}=\lim_{l\rightarrow \infty}[E\oplus \mR^l,\mR^{k+l}]_{\mathrm{prop}}.
$$
Recall that the stable homotopy and cohomotopy groups of a space $X$ are similarly defined
$$
\pi_k^S(X)=\lim_{l\rightarrow \infty}\langle S^l S^k,S^l X\rangle\quad \text{and}\quad \pi_S^k(X)=\lim_{l\rightarrow \infty}\langle S^l X,S^l S^k\rangle.
$$
A direct corollary of Theorem~\ref{thm:iso} is then
 \begin{corollary}
 \label{cor:stable}
 Let $M$ be a finite $CW$ complex of dimension $m$ and $E$ a vector bundle of rank $n$. Then $Q$ induces a bijection $[E,\mR^k]^S_{\mathrm{prop}}$ with $\pi^k_S(\Th(E))
 \end{corollary}

Stable homotopy and cohomotopy groups are related via Spanier-Whitehead duality, which we recall now. We refer to the original references~\cite{Spanier:1956hv,Spanier:1955hg} for these statements. Let $i:X\rightarrow S^N$ be a sufficiently nice embedding of a sufficiently nice space $X$ into a sphere (e.g. a smooth embedding of a compact manifold, or the inclusion of a CW complex as a subcomplex). Then the space $D_NX=S^N\setminus i(X)$ is a Spanier-Whitehead dual of $X$. The stable homotopy type of $D_NX$ is well defined: It is independent of the dimension $N$ and the choice of embedding. The fundamental result is that $\lim_{l\rightarrow \infty}[S^lX,S^lY]$ is in bijection with $\lim_{l\rightarrow \infty}[S^lD_NY,S^lD_NX]$. In particular, the stable cohomotopy groups of $X$ are the stable homotopy groups of $D_NX$ with a dimension shift. Now let us assume that $M$ is a compact manifold with boundary $\partial M$. There is a unique (up to isotopy) embedding of $M$ into $\mR^{m+n}$ for $n$ sufficiently large. Let $E$ be the normal bundle of such an embedding, i.e. let $E$ be the stable normal bundle of $M$. Atiyah~\cite[Proposition 3.2]{Atiyah:1961dp} showed that $SD_{m+n}(M/\partial M)\simeq\Th(E)$. If the boundary $\partial M$ is empty we should interpret $M/\partial M$ as $M$ with a disjoint basepoint added. The Spanier-Whitehead dual of a sphere is $D_{n+m} S^{n+m-k}=S^{k-1}$. We have
\begin{align*}
\pi_{n+m-k}^S(M/\partial M):&=\lim_{l\rightarrow\infty}\langle S^lS^{n+m-k},S^l M/\partial M\rangle\\
&=\lim_{l\rightarrow \infty}\langle S^l D_{n+m}(M/\partial M),S^l D_{n+m} S^{n+m-k}\rangle\\
&=\lim_{l\rightarrow\infty}\langle S^{l-1}\Th(E),S^l S^{k-1}\rangle\\
&=\lim_{l\rightarrow \infty}\langle S^l\Th(E),S^l S^k\rangle\\
&=\pi^k_S(\Th(E)).
\end{align*}
Theorem~\ref{thm:iso} states that $\pi^k_S(\Th(E))$ is in bijection with $[E,\mR^k]^S_{\mathrm{prop}}$. We have proven Theorem~\ref{thm:atiyah}.

\section{Framed submanifolds and cobordisms}
\label{framed}
Pontryagin~\cite{Pontryagin:1959vj} showed that homotopy classes of maps $M\rightarrow S^k$, where $M$ is a closed manifold, are in one to one correspondence with framed cobordism classes of $(n-k)$-dimensional manifolds in $M$. Framed cobordism classes are also invariants of homotopy classes of proper maps $E\rightarrow \mR^k$ but they are not complete, nor is every cobordism classed realized by some proper map. In this section we discuss this.

\subsection{Invariants of proper maps: framed submanifolds and cobordisms}
Let $M$ be a smooth $m$-dimensional manifold and $N$ be a connected oriented smooth $k$-dimensional manifold. Every continuous proper map is homotopic as a proper map to a smooth proper map, hence we can consider only proper smooth maps in the proper homotopy classification of proper maps. Suppose that $f:M\rightarrow N$ is a smooth map that is proper. Proper maps between manifolds are closed maps. The set of critical points is closed, hence the set of regular values of a proper map is open. By Sard/Brown's theorem regular values of $f$ are residual, and by Baire's category theorem it follows that the regular values are dense. Thus the set of regular values of a proper map is open and dense. An example of smooth function $\mR_{>0}\rightarrow \mR$ whose set of regular values is not open is $x\mapsto \frac{1}{x(2+\sin(x))}$, but of course this map fails to be proper and closed. The preimage of a regular value $y$ is a closed submanifold $X=f^{-1}(y)$ of dimension $m-k$. Such a manifold can be \emph{framed}: Let $e_1,\ldots,e_k$ be a basis of $T_yN$ that is compatible with the orientation of $N$. Then for every $x\in X$ the differential of $f$ induces an isomorphism $df_x:N_xX\rightarrow T_yN$ of the normal space $N_xX$ to $X$ at $x$ with $T_yN$. Then $(\nu_f)_x=((df_x)^{-1}(e_1),\ldots, (df_x)^{-1}(e_k))$ is an ordered basis of the normal space $N_xX$ at $x$. Letting $x$ vary, this patches together to a map $\nu_f$ that trivializes the normal bundle of $X$. The map $\nu_f$ is called the framing of $X$. We call $(X,\nu_f)$ a Pontryagin manifold of $f$ and it depends on the choices we made.

Let $F:[0,1]\times M\rightarrow N$ be a homotopy of proper maps between $f_0=F(0,\cdot)$ and $f_1=F_1(1,\cdot)$. By a reparametrization of the homotopy variable, we may assume that $F(t,x)=f_0(x)$ and $F(1-t,x)=f_1(x)$ for $t$ small. If $y$ is a regular value of the maps $F,f_0$ and $f_1$ simultaneously then $(W=F^{-1}(y),\nu_F)$ is a framed compact submanifold with framed boundary $(X_0=f_0^{-1}(y),\nu_{f_0})$ and $(X_1=f_0^{-1}(y),\nu_{f_1})$. The framed manifold $(W,\nu_F)$ is a framed cobordism between the framed manifolds $(X_0,\nu_{f_0})$ and $(X_1,\nu_{f_1})$.
Being framed cobordant defines an equivalence relation on the set of framed submanifolds and the framed cobordism class of a Pontryagin manifold of a proper map $f:M\rightarrow N$ does not depend on the choice of regular value $y$ and the choice of oriented basis of $T_yN$ and is an invariant of the proper homotopy class of $f$. We denote the set of framed $(m-k)$-dimensional closed submanifolds of $M$ up to framed cobordism by $\Omega^{\mathrm{fr}}_{m-k}(M)$.

\subsection{The Pontryagin-Thom construction}

The framed cobordism class of the preimage of a regular value is in some cases enough to recover the homotopy class of the map: Suppose $M$ is closed and $(X,\nu)$ is a $(m-k)$-dimensional framed submanifold. Out of this data we can construct a (proper) map $f:M\rightarrow S^k$, for which $(X,\nu)$ is a Pontryagin manifold: We define $f$ to map $X$ to the northpole $y$ of $S^k$ and describe what happens in a tubular neighborhood of $X$. The framing $\nu$ defines, for each point $x\in X$ a diffeomorphism of the normal space around $x$ to a neighborhood of $y$. We use this to extend the map to the tubular neighborhood of $X$ in $M$. One can arrange this in such a way that if one approaches the boundary of the tubular neighborhood, the image under $f$ converges to the south pole. The map $f$ can now be extended to the whole of $M$ by mapping everything outside the tubular neighborhood to the south pole. The northpole is a regular value for $f$ and the Pontryagin manifold at the north pole is exactly the framed manifold $(X,\nu)$. This construction also works for framed cobordisms. This proves the following theorem.

\begin{theorem}
The Pontryagin-Thom construction gives a one to one correspondence between the set $\Omega^{\mathrm{fr}}_{m-k}(M)$ of framed cobordisms in $M$ and the set $ [M,S^k]$ of homotopy classes of maps from $M$ to $S^k$.
\end{theorem}

For more details of the Pontryagin-Thom construction in this classical setting we refer to Milnor~\cite{Milnor:_FIovTN9} and Pontryagin~\cite{Pontryagin:1959vj}.

\subsection{How good of an invariant is the framed cobordism class of the Pontryagin manifold of a proper map?}
\label{sec:counterexample}
A proper map $f:\mR^n\rightarrow \mR^k$ is proper homotopic to a map $Pg$, where $g:S^{n-1}\rightarrow S^{k-1}$. Let $y\in S^{n-1}$ be a regular value of $g$, and $(X,\nu_g)$ be the Pontryagin manifold of $g$ at $y$. The value $y\in S^{k-1}\subseteq \mR^k$ is also a regular for the map $Pg$. The Pontryagin manifold of $Pg$ at $x$ is $(X,(\nu_{g},\mu))$, where $\mu$ is the last component of the framing which points radially outward from the sphere. So a framed submanifold cannot occur as a Pontryagin manifold if it is not framed cobordant to a framed submanifold that lies on a sphere where the last component of the framing is radially pointing outward.

Let us discuss an explicit example of a framed manifold that does not occur as the Pontryagin manifold of a map. Consider the submanifold $X=\{-1,1\}\subseteq \mR$ with framing $\nu_{-1}=\nu_1=\frac{\partial}{\partial t}$. Then $(X,\nu)$ cannot occur as the preimage of a regular value of a proper map $f:\mR\rightarrow \mR$. Suppose on the contrary that such a map exists with $f(-1)=f(1)$. From the framing and Taylor's theorem we see that there exists an $\epsilon>0$ such that $f(-1+\epsilon)>f(-1)$ and $f(1-\epsilon)<f(1)$. The intermediate value theorem then gives the existence of another point $-1+\epsilon<p<1-\epsilon$ such that $f(p)=f(-1)=f(1)$. Hence $f^{-1}(f(1))\not=X$ and we conclude that there does not exist a proper $f:\mR\rightarrow \mR$ with $(X,\nu)$ as a Pontryagin manifold.

But there are also framed submanifolds which are framed cobordant to a framed submanifold which is contained in the unit sphere and has a framing with last component pointing radially outward which do not arise from proper maps. To see this, consider the manifold $X=\{-2,-1,1,2\}$ with framing $\nu(-2)=\nu(2)=\frac{\partial}{\partial t}$ and $\nu(-1)=\nu(1)=-\frac{\partial}{\partial t}$. Then $(X,\nu)$ is framed cobordant to the empty set, however it cannot occur as the Pontryagin manifold of a proper map: If $y$ is the regular value for which $(X,\nu)$ is hypothetically the Pontryagin manifold at $y$, there must be a point $x \in (-1,1)$ such that $f(x)=y$, by the same reasoning as above.

Finally we discuss the fact that the invariant is not complete. The maps $f,g:\mR\rightarrow \mR$ given by $f(x)=x^2$ and $g(x)=-x^2$ are not proper homotopic. However as the maps are not surjective the framed cobordism class of both maps is the empty manifold. Hence the framed cobordism class of a regular value cannot distinguish these maps. Here is a more complicated example: Let $f,g:S^3\rightarrow S^2$ be the Hopf map and the Hopf map precomposed with a degree $-1$ map of $S^3$ respectively. These maps represent $+1$ and $-1$ in $\pi_3(S^2)\cong \mZ$ and are not homotopic. By Theorem~\ref{maintheorem} $Pf$ and $Pg$ are therefore not proper homotopic, however their Pontryagin manifolds are framed cobordant. To see this note that $QPf$ and $QPg$ are the suspensions of $f$ and $g$ and the Pontryagin manifolds of $QPf$ and $QPg$ can be identified with those of $Pf$ and $Pg$. The suspension map $S:\pi_3(S^2)\cong \mZ\rightarrow \pi_4(S^3)\cong \mZ/2\mZ$ is the  reduction modulo $2$. The maps $QPf$ and $QPg$ are homotopic, so their Pontryagin manifolds must be framed cobordant. But this implies that the Pontryagin manifolds of $PF$ and $PG$ are framed cobordant by general position.

\subsection{Open manifolds and the Pontryagin-Thom construction}

\label{sec:speculation}
In Corollary~\ref{cor:stable} we have seen that the homotopy classes of proper maps out of vector bundles stabilizes. We expect that if $M$ is an arbitrary open manifold the homotopy classes of proper maps $[M\times \mR^l,\mR^{k+l}]_{\mathrm{prop}}$ stabilizes when $l\rightarrow \infty$. This suggests that there is a stable Pontryagin-Thom construction for proper maps. A framed submanifold $(X,\nu)\in \Omega^{\mathrm{fr}}_{m-k}(M)$ gives rise to a framed submanifold $((X,0),\nu\oplus \mu)\in \Omega^{\mathrm{fr}}_{m-k}(M\times \mR^l)$ via stabilization. The framing $\nu\oplus \mu $ extends the framing $\nu$ with a fixed basis $\mu$ of $\mR^l$. Since we expect that the homotopy classes of proper maps stabilizes we also expect that there is a well defined stable bijective Pontryagin-Thom construction $\Omega^{\mathrm{fr}}_{m-k}(M\times \mR^l)\rightarrow [M\times \mR^l,\mR^{l+k}]_{\mathrm{prop}}$ for $l$ sufficiently large.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


text:
sasquatch chronicles blog
Upcoming Show: The William Roe Encounter
I, W. Roe of the City of Edmonton, in the province of Alberta make oath and say,
(1) That the exhibit A attached to this, my affidavit, is absolutely true and correct in all details.
Sworn before me in the City of Edmonton, Province of Alberta, this 26th day of August, A.D. 1957.
(Signed) William Roe
(Signed) by W.H. Clark
Assistant Claims Agent
Number D.D. 2822
EXHIBIT A.
Ever since I was a small boy back in the forest of Michigan, I have studied the lives and habits of wild animals. Later, when I supported my family in Northern Alberta by hunting and trapping, I spent many hours just observing the wild things. They fascinated me. But the most incredible experience I ever had with a wild creature occurred near a little town called Tete Jaune Cache, British Columbia, about eighty miles west of Jasper, Alberta.
I had been working on the highway near Tete Jaune Cache for about two years. In October, 1955, I decided to climb five miles up Mica Mountain to an old deserted mine, just for something to do. I came in sight of the mine about three o'clock in the afternoon after an easy climb. I had just come out of a patch of low brush into a clearing when I saw what I thought was a grizzly bear, in the bush on the other side. I had shot a grizzly near that spot the year before. This one was only about 75 yards away, but I didn't want to shoot it, for I had no way of getting it out. So I sat down on a small rock and watched, my rifle in my hands.
I could see part of the animal's head and the top of one shoulder. A moment later it raised up and stepped out into the opening. Then I saw it was not a bear.
This, to the best of my recollection, is what the creature looked like and how it acted as it came across the clearing directly toward me. My first impression was of a huge man, about six feet tall, almost three feet wide, and probably weighing somewhere near three hundred pounds. It was covered from head to foot with dark brown silver-tipped hair. But as it came closer I saw by its breasts that it was female.
And yet, its torso was not curved like a female's. Its broad frame was straight from shoulder to hip. Its arms were much thicker than a man's arms, and longer, reaching almost to its knees. Its feet were broader proportionately than a man's, about five inches wide at the front and tapering to much thinner heels. When it walked it placed the heel of its foot down first, and I could see the grey-brown skin or hide on the soles of its feet.
It came to the edge of the bush I was hiding in, within twenty feet of me, and squatted down on its haunches. Reaching out its hands it pulled the branches of bushes toward it and stripped the leaves with its teeth. Its lips curled flexibly around the leaves as it ate. I was close enough to see that its teeth were white and even.
The shape of this creature's head somewhat resembled a Negro's. The head was higher at the back than at the front. The nose was broad and flat. The lips and chin protruded farther than its nose. But the hair that covered it, leaving bare only the parts of its face around the mouth, nose and ears, made it resemble an animal as much as a human. None of this hair, even on the back of its head, was longer than an inch, and that on its face was much shorter. Its ears were shaped like a human's ears. But its eyes were small and black like a bear's. And its neck also was unhuman. Thicker and shorter than any man's I had ever seen.
As I watched this creature, I wondered if some movie company was making a film at this place and that what I saw was an actor, made up to look partly human and partly animal. But as I observed it more, I decided it would be impossible to fake such a specimen. Anyway, I learned later there was no such company near that area. Nor, in fact, did anyone live up Mica Mountain, according to the people who lived in Tete Jaune Cache.
Finally the wild thing must have got my scent, for it looked directly at me through an opening in the brush. A look of amazement crossed its face. It looked so comical at the moment I had to grin. Still in a crouched position, it backed up three or four short steps, then straightened up to its full height and started to walk rapidly back the way it had come. For a moment it watched me over its shoulder as it went, not exactly afraid, but as though it wanted no contact with anything strange.
The thought came to me that if I shot it, I would possibly have a specimen of great interest to scientists the world over. I had heard stories of the Sasquatch, the giant hairy Indians that live in the legends of British Columbia Indians, and also many claim, are still in fact alive today. Maybe this was a Sasquatch, I told myself.
I levelled my rifle. The creature was still walking rapidly away, again turning its head to look in my direction. I lowered the rifle. Although I have called the creature "it", I felt now that it was a human being and I knew I would never forgive myself if I killed it.
Just as it came to the other patch of brush it threw its head back and made a peculiar noise that seemed to be half laugh and half language, and which I can only describe as a kind of a whinny. Then it walked from the small brush into a stand of lodgepole pine.
I stepped out into the opening and looked across a small ridge just beyond the pine to see if I could see it again. It came out on the ridge a couple of hundred yards away from me, tipped its head back again, and again emitted the only sound I had heard it make, but what this half- laugh, half-language was meant to convey, I do not know. It disappeared then, and I never saw it again.
I wanted to find out if it lived on vegetation entirely or ate meat as well, so I went down and looked for signs. I found it in five different places, and although I examined it thoroughly, could find no hair or shells of bugs or insects. So I believe it was strictly a vegetarian.
I found one place where it had slept for a couple of nights under a tree. Now, the nights were cool up the mountain, at this time of year especially, and yet it had not used a fire. I found no sign that it possessed even the simplest of tools. Nor a single companion while in this place.
Whether this was a Sasquatch I do not know. It will always remain a mystery to me, unless another one is found.
I hereby declare the above statement to be in every part true, to the best of my powers of observation and recollection.
6 Responses to "Upcoming Show: The William Roe Encounter"
Annette H
Looking forward to this. Maybe you should call it the Sasquatch Confessionals.
July 11th, 2019 Log in to Reply
Melissa K
I saw a documentary with his daughter and she talked about how she drew this picture for him as he described it to her.
I think John Green was in it and talked about interviewing Roe, too.
Denise F
Charles R
One of the few encounters where an ear is seen as the hair usually covers it. At 6 foot, maybe 300 lbs this was most likely a very young female. She walked away in the same fashion as did Patty and had the same body frame and also facial hair. I am glad my wife does not have facial hair. It is amazing that being so far away in the wilderness, this creature did not seem to frighten him, and he even went to look for signs of her travels. I suspect if he had come this close to a full grown male, he might not have chosen this option.
Lol…great promo video Wes.
LOL, you dat! Dang Wes!
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If the API is connected to "backend A", the "app 2.0" that is being reviewed does not work properly.
If the API is connected to "backend B" ⇒ "App 1.0" in the public does not work properly, there is no "app 2.0" in the store.
Can you give me advice on how to operate?
In the case of what to do during review, you can mention those same details via that app's meta data note field.
As well, a short demo video that the review team can check w/backend B in action can spare you from juggling and upsetting app 1.0 users...simply upload that video where review can find it, adding a link to the note. The demo video has the added benefit of working as an app 2.0 Preview video in the store via meta data.
Easy solution. Run backend A and backend B. In fact, you don't have any option. No one is obligated to update. If you break app 1.0, they will leave 1-star reviews of your app and you won't have many users of app 2.0.
(For the reasons stated by john daniel.....) If I understand your problem, you need to create new APIs that reference Backend B and use only those new APIs in App 2.0. At the same time you need to maintain the old APIs that are used by App 1.0 and reference backend A.


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Q: Is there a way to compile a solution from a unit test and fail the test if the compile fails? I want to overwrite the Program.cs file in a solution I have created for a test, and then compile that solution - as a unit test.
With our library we have in the designer that comes with it a button titled Generate Code. When you click that button it pops up with the code for a sample Program.cs file that calls our library specifically written for the template created in the designer.
We want to have a unit test that verifies that the code we created compiles. We're not testing that it runs, just that it compiles. So the thought is we create a project, replace the Program.cs file in the project, and then compile the project. Successful compile, the test passes. Compile fails, the test fails.
What's driving this is we had a mistake in the code where it had "${name}.compile()" instead of $"{name}.compile()" - and that will easily be found with this.
So, can it be done?

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