Stratifications and Cohomology Computations 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!
 A: 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.)
A: 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. 
A: 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
A: 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 $k$.
