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The examples mentioned so far, made the point that Cech cohomology differs from singular cohomology in many examples which are, to the eyes of most topologists, quite pathological. … I want to concentrate on an example of an ordinary homology theory, which is probably much less well known than Cech cohomology, but quite interesting in geometric contexts: stratifold homology (see Kreck's …

While for the tensor algebra instead, one could compute its group cohomology via a Künneth theorem [edit: as Thorsten Ekedahl pointed out, this computes only the cohomology of $G\times G$], I have not … So my question is: Is there a general procedure for computing the cohomology groups $H^i(G; Sym(V))$ (with $H^i(G; V)$ as input)? …

To a local system one can associate (singular) cohomology groups with local coefficients. … I guess, these cohomology groups with local coefficients should be isomorphic to your ''vector bundle deRham cohomology'', though I know no reference at the moment. …

I take the liberty to reference to a paper of mine (which is hopefully no bad style): Spectral Sequences in String Topology. Here, the K-homology of the free loop spaces of spheres and complex project …

an $A_\infty$-ring spectrum (e.g. if $E =ko, ku, KO, KU, TMF, MO, MSO, MSpin \dots$) [EKMM, IV.4] $E$ is even and Landweber exact (e.g. if $E = MU, E(n), E_n, \dots$) [Adams' lectures on generalized cohomology … to Ext^1_{\mathbb{Z}}(E_{k-1}X, \mathbb{Z}) \to (IE)^kX \to Hom_{\mathbb{Z}}(E_kX, \mathbb{Z}) \to 0.$$ That means that there exists always a short exact sequence computing from $E$-homology the $IE$-cohomology …

Given a Morse function on a Riemannian smooth manifold, you obtain (under mild conditions) a chain complex computing its homology, the Morse complex. Its generators are the critical points and differe …

There are some applications of topology/cohomology to combinatorics and combinatoric geometry. … There are many other results in convex geometry/polytope theory which use topological methods and, in particular, cohomology. …

1) If you want to compute the rational or real cohomology of something, you can try to use rational homotopy theory. … space. 2) If you can show that your space is a $BG$, its cohomology equals the group cohomology of $G$ which is computable in some cases. …