Questions tagged [cohomology]
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
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Can anyone give me a good example of two interestingly different ordinary cohomology theories?
An answer to the following question would clarify my understanding of what a cohomology theory is. I know it's something that satisfies the Eilenberg-Steenrod axioms, and I know that those axioms ...
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Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?
Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
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Are generalized cohomology theories a homotopy category of some category of invariants?
I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
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Cohomology and fundamental classes
Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
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Representablity of Cohomology Ring
I know that the individual cohomology groups are representable in the homotopy category of spaces by the Eilenberg-MacLane spaces. Is it also true that the entire cohomology ring is representable? If ...
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Commutativity in K-theory and cohomology
The Chern classes give a map $f : BU \to \prod_n K(\mathbb{Z},2n)$, which is a rational equivalence. However, it is not an equivalence over $\mathbb{Z}$ because the cohomology of $BU$ is just a ...
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How to compute the (co)homology of orbit spaces (when the action is not free)?
Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
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References for syntomic cohomology
Could anyone point to good readable references for learning about syntomic cohomology?
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Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
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What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?
I have a few elementary questions about cup-products.
Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an ...
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What is (co)homology, and how does a beginner gain intuition about it?
This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
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Non-zero sheaf cohomology
Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...
