# Tagged Questions

Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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### path to (co)homology

I want to understand (co)homology, but in the research I've done, it seems as though despite being 'birthed' within algebraic topology, its usefulness has allowed it to permeate throughout all ...
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### Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
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### A doubt in Whitehead's proof about cohomology with local coefficients [closed]

In the proof of Theorem 4.9 says that $p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$ has image $Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$. ...
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### Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
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### Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
### What is the definition for the subgroup $Z^p(K;G)$ in cohomology? [closed]
I am self-studying cohomology. And I figured that the kernel of a cohomology group $H^p(K;Z)$ called $Z^p(K;Z)$ cocycle subgroup is a set when the p-coboundary $d$ operator acts on $C^p(K)$ a p-...