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Results tagged with cohomology
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user 3634
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, ...
25
votes
Accepted
Image of a map on cohomology rings
No. Consider the Hopf map $\eta:S^3\to S^2$. If there were such a space $Z$, it would have $\widetilde H^*(Z)=0$, so at the very least $Z$ would be stably trivial, forcing $\eta$ to be stably trivial; …
- 12.5k
19
votes
4
answers
2k
views
Difference between represented and singular cohomology?
Ordinary cohomology on CW complexes is determined by the coefficients. … There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or
by defining $\widetilde H^n(X;G) = [X, K(G,n)]$. …
- 1,243
14
votes
Teaching Steenrod Operations
I like to observe that the diagonal map $X\to X\times X$
is $\mathbb{Z}/2$-equivariant, hence induces a map of homotopy colimits.
Analyzing this map with $X = K( \mathbb{Z}/2,n)$ gives the squares.
Th …
- 12.5k
4
votes
Accepted
Homology groups of compact subset of $\mathbb{R}^2$
This is answered in the paper
The singular homology group of planar sets do not behave anomalously
by Andreas Zastrow
This appears to be a link to the paper: http://at.yorku.ca/i/d/e/b/11.ht …
- 12.5k
8
votes
2
answers
2k
views
Splitting of the Universal Coefficients sequence
to my taste,
is to use the fibration sequence $K(\mathbb{Z}, n) \to K(\mathbb{Z}, n) \to
K(\mathbb{Z}/k, n)$ (I'm using $\mathbb{Z}/k$ coefficients for simplicity)
to give a long exact sequence of cohomology …
- 3,526