Cohomology Of Classifying Space

In recent times, cohomology of classifying space has become increasingly relevant in various contexts. algebraic topology - Why is (co)homology useful and in which way .... For instance, I heard that cohomology was used in the proof of the Weil conjectures, and I also heard that there's a cohomology theory (Group cohomology) which can be used in group theory. I just want to get a clue at why (co)homology is useful in these areas and how it is applied there.

So what is Cohomology? - Mathematics Stack Exchange. There are many versions of cohomology which all use the same basic approach, but the most intuitive version for someone who has gone through the usual calculus sequence along with linear algebra and some basic analysis is de Rham cohomology. Homology came first and its goal/description is pretty much the same. algebraic topology - Difference between Homology and Cohomology .... If $\phi$ induces homology isomorphisms in all dimensions, then $\phi$ induces cohomology isomorphism in all dimensions.

That is to say, if we can not use homology to distinguish two space, so do cohomology. However my teacher told us that the essential difference between them is the ring structure on the cohomology. algebraic topology - Intuitive Approach to de Rham Cohomology .... Moreover, on the other hand, the definition of de Rham cohomology always comes unprovided with such intuitive approach.

My question is: how may de Rham cohomology be intuitively understood? How is de Rham cohomology useful? From another angle, to that end, the answer to the question of what the de Rham cohomology groups are good for might be that they're a particularly useful and concrete way of defining and computing the cohomology of a space.

There are multiple cohomology theories (not all of which are well-defined for all spaces): singular cohomology, cellular cohomology, etc. Sheaf cohomology: what is it and where can I learn it?. Incidentally, sheaf cohomology provides a very simple proof that de Rham cohomology agrees with ordinary cohomology (at least when you agree that ordinary cohomology is cohomology of the constant sheaf, here $\mathbb {R}$) because the de Rham resolution is a soft resolution of the constant sheaf $\mathbb {R}$, and you can thus use it to compute ... Why homology with coefficients?

In this context, finally, (co)homology group with coefficients in a field with char 0 have different analytical description ('de Rham cohomology' etc) and (at least in some situations) some extra structue (e.g. 'Hodge structures' in cohomology of projective complex manifolds). Computing the homology and cohomology of connected sum. The procedure for finding homology and cohomology of the spaces in question is a neat little trick.

From here on out, I'll just treat the homology case, but the cohomology follows from the same arguments. In this context, comparison morphism between Čech and sheaf cohomology. Then you can “play” with the “grid”, by moving “along the diagonal” from the complex computing the Cech cohomology to the (orthogonal) one computing sheaf cohomology. There’s probably a cleaner way with spectral sequences if you know them.

integral cohomology ring of real projective space. 7 The ring structure of the integral cohomology can be determined by the ring structure of the mod 2 cohomology.