## References/surveys concerning characteristic classes of flat vector bundles

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I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}_n(\mathbb{R})$ with the discrete topology, orientable flat $n$-dimensional real vector bundles are classified by $BG$, so the characteristic classes I'm looking for are elements of the cohomology of $BG$ with respect to various systems of coefficients.

The one source I know of is Morita's book "The geometry of characteristic classes", but this is not very comprehensive or up-to-date. I also have found many references about Chern-Simons invariants, but they all seem to be written from the perspective of algebraic geometry or mathematical physics or differential geometry. I'd really like a source that is as topological as possible; for instance, for ordinary characteristic classes I prefer the approach taken in Milnor-Stashef to the approach via Chern-Weil theory. But any sources at all are welcome.

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## 2 Answers

I am not aware of much recent activity (after 2001, when Morita wrote his book) in this field, so I think it is still very valuable. Chern-Simons theory means something rather different today. The useful information on characteristic classes of flat vector bundles (CCFVB) seems to be scattered throughout the literature; one of the reasons might be that there is no complete picture known. Googling gives you a lot of sources, but I completely agree that a modern detailed and comprehensive exposition is missed.

Also, as far as I know, Chern-Weil theory is an indispensible ingredient of the theory. Purely topological methods are not very sensitive to flat bundles while Chern-Weil theory allows you to import methods from Lie algebra cohomology etc.

Here is a list of texts from which I learnt something on the subject:

Karoubi: Homologie cyclique et K-Theorie algebrique.

Jones, Westbury: Homology spheres, eta-invariants and algebraic K-theory. The first few sections form an informative survey which is worth reading even if you are willing to ignore the main problem they consider.

Dupont: Curvature and characteristic classes (the last section)

Kamber, Tondeur: Foliated bundles and characteristic classes

A modern approach to CCFVB is to use differential cohomology (or differential characters). If you google, you find more information on that.

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I think the names of Johan Louis Dupont ( http://home.imf.au.dk/dupont/ ), Chih-Han Sah (1934--1997) (SUNY Stony Brook) are appropriate. Dupont's homepage contains related articles e.g.

Homology of O(n) and O^1(1,n) made discrete; an application of edgewise subdivision by J. Dupont, M. Bökstedt and Morten Brun, J. Pure Appl. Algebra, 123 (1998), 131-152.

and many of his papers are in arXiv, e.g.

Regulators and characteristic classes of flat bundles Johan Dupont, Richard Hain, Steven Zucker http://arxiv.org/abs/alg-geom/9202023

The Lie groups made discrete arise in algebraic K-theory, for small n the groups K_n are related to "scissors congruence groups" (as far as I understand), see let me quote http://reh.math.uni-duesseldorf.de/~topologie/scissors/

"The basic reference for the school is the monograph [D] J.L. Dupont, Scissors congruences, group homology and characteristic classes, World Scientific.

Further references occuring below are

[M] Milnor, On the homology of Lie groups made discrete. Commentarii Mathematici Helvetici, Vol. 58, No. 1, 72--85, 1983

[S] Suslin, A.A., Algebraic K-theory of fields. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222--244, Amer. Math. Soc., Providence, RI, 1987"

And further quote:

"Scissors congruences

Two polytopes in euclidean n-space are called scissors congruent if they can be subdivided into finitely many pieces such that each piece in the first polytope is congruent to exactly one piece in the second polytope.

Elementary geometric considerations show that polytopes in the plane are scissors congruent if and only if they have the same area. Hilbert's 3rd problem was the question whether volume determines the scissors congruence class also in 3-space. The answer was given by Max Dehn almost immediately: In 1900, he described an invariant with values in R ⊗Z R/Z which shows that the answer is no. Only 1965 J. P. Sydler proved that volume and Dehn invariant together determine the scissors congruence class in 3-space. Higher dimensional analogues are still unsolved. There are variants for spherical and hyperbolic geometry, which are open even in dimension 3.

From a modern point of view, these classical questions are closely related to the computation of the homology of Lie groups considered as discrete groups. Furthermore there are interesting connections to deep questions about the algebraic K-theory of the complex numbers."

See also:

http://en.wikipedia.org/wiki/Hilbert%27s_third_problem

P. Cartier, Décomposition des polyèdres : le point sur le troisième problème de Hilbert, Séminaire Bourbaki, 1984-85, n° 646, p. 261—288.

Hilbert’s 3rd problem and Invariants of 3-manifolds by Walter Neumann, G&T Monographs, 1998.

http://www.emis.de/journals/GT/GTMon1/paper19.abs.html

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