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I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the subject matter. Also, my stratification consists of orbits of an action of a connected, simply-connected complex semisimple Lie group, so I would also appreciate references and suggestions relevant to my situation.

Thanks!

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I think you are looking for Spectral Sequences (which thought makes me feel somewhat gloomy...) on which there is a thorough reference by McCleary. Good luck to you! –  some guy on the street Apr 23 '13 at 20:49

4 Answers 4

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As some guy on the street hinted, any "elementary" inductive approach is a spectral sequence in disguise. You have not indicated what topological invariants you have in mind. If Betti numbers suffices for your needs, then in some instances the spectral sequences become relatively simple.

One such instance comes from the stratification of Grassmannians (or more generally flag manifolds) by Schubert cells. In this case the $k$-th Betti number of the corresponding f;ag space is equal to the number of Schubert cells of dimension $k$.

More generally, suppose that you have a Whitney stratification of a compact space $X$ with the following properties.

  • All the strata are diffeomorphic to open balls.
  • There exists no pair of strata $(S,S')$ such that $|\dim S-\dim S'|=1$.

Then the $k$-th Betti number of $X$ is equal to the number of strata of dimension $k$.

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@Liviu: You have the get out word "elementary"! My problem with exact and spectral sequences has long been determining the possible extensions, since that was what my thesis was about, for function spaces. So I was quite surprised later to get with a groupoid van Kampen theorem precise colimit results involving consecutive dimensions. Analysing the reasons of this success was one of the intuitions behind the book referred to in my answer, namely the intuition of requiring invariants with structure in a range of dimensions, groupoids having structure in dimensions $0$ and $1$. So, ... –  Ronnie Brown Apr 24 '13 at 21:19

I don't know if this is the direction you're interested in, but the book "An Introduction to Intersection Homology Theory" by Kirwan and Woolf is a nice readable book that has a lot about stratifications and their connection to topological invariants. The last section of the book deals with the particular case of the flag variety for a semisimple Lie group (and the famous Beilinson-Bernstein correspondence), so that might be helpful. (You may also want to look at the book "D-modules, perverse sheaves, and representation theory" by Hotta, Takeuchi and Tanisaki that focuses on the connection to Lie groups, their representations, and homogeneous spaces.)

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Is the stratification you consider an example of a filtered space, i.e. a space $X$ and a sequence $X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots $ of subspaces? The algebric topology of these is considered in the book published by the EMS in 2011 as Tract in Mathermstics Vol 15, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. There are also comments about stratifications in Grothendieck's "Esquisse d'un programme" Section 5, which may be relevant, of which a translation is published in a book by Leila Schnepp.

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There is the Atiyah-Jones conjecture (now theorem). It even has its own (brief) Wiki page

http://en.wikipedia.org/wiki/Atiyah%E2%80%93Jones_conjecture

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