2 Fixed title

1

# Correspondeces in Topology

I had one or two little fights with correspondences in the context of algebraic geometry where an elementary correspondence $C:X\to Y$ of connected smooth $k$-Schemes seems to be defined as an irreducible closed (reduced) subscheme $C\hookrightarrow Y\times_k X$, such that the projection to $X$ is finite an surjective.

Further, I have vaguely heard of correspondences in topology (invented by Lefschetz?) where it seems to me, that such a thing is a cohomology class in $H^*(Y\times X)$ for compact, oriented manifolds `$X,Y$`. Using Poincare duality and the cohomology pushforward functor `$(-)_!$` I can associate a cohomology class `$(\Delta_f)_!(1)$` in `$H^*(Y\times X)$` to a map $X\to Y$.

My questions are:

1. I can not really see the analogy of the concepts, except that in booth cases one can associate a corresponces to a morphism by its graph. So, what (or how deep) is the analogy?

2. I know (only a very few) applications of correspondences in algebraic geometric, but of none in topology. What are they good for? Where can I find applications?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9589897394180298, "perplexity_flag": "middle"}