# Questions tagged [homology-cohomology]

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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### Group cohomology reference

I'm interested in studying group cohomology (for discrete groups). Are there accessible (lecture) notes that give a nice overview of the basics for group cohomology and develop the categorical ...
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### Why is $X \mapsto hom(\pi_*^{st}X, \mathbb{Q})$ the same as ordinary rational cohomology?

I am trying to understand the notion of Anderson duality from appendix B of this paper https://arxiv.org/abs/math/0211216 by Hopkins and Singer. But I somehow get stuck at the very first steps. I am a ...
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### example of Grade of module [duplicate]

Let $R$ be a commutative Noetherian ring, $I$ be a proper ideal of $R$ and $M$ is finitely generated $R$-module. I want to find an example of module $M$ such that $$grade(I,M)>grade(I,R)$$ thank ...
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### The first homology group $H_1(M)$ of a compact manifold $M$ is always $\mathbb Z_i \oplus \mathbb Z_j\oplus...\oplus\mathbb Z_n$ ($i,j,...,n\ge 2$)?

The dimension of the first homology group $H_1(M)$ is the number of (nonequivalent) loops of the manifold $M$. For all the cases I know, such as $RP^n$, $S^n$, cylinder, n-torus, klein bottle and so ...
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### Relation between mod 2 Betti numbers and integral cohomology

Let $M$ be an orientable connected manifold of finite type. My question is that if $H^{i}(M,\mathbb{Z}_{2})$ is non-zero, then can we say that $H^{i}(M,\mathbb{Z})$ is non-zero? I know that this is ...
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### “Usual argument for a bicomplex” in order to extend a cocycle

I was reading this article and I got stuck in the proof of Proposition 2.1. In particular I don't get the last paragraph of the proof when it talks about extending to a cocycle and using a similar ...
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### Homology and homotopy groups and holes

Q. In what sense do homology and homotopy groups "count" or "detect" holes? And when do they differ in their hole counting? I'm seeking a high-level view of these questions, ...
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### Pullback of Lie derivative acting on $k-$ forms

I have to prove the following. Let $M$ be a differentiable smooth manifold and let $\chi \in \Gamma(TM)$ a smooth vector field on $M$. Denote by $\mathcal{L}_{\chi}$ the Lie derivative along $\chi$ ...
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### General Applications of Persistent Homology [closed]

Are there any general applications of persistent homology (or topological data analysis in general) ? . All the applications I seem to find relate to very specific applications so I was wondering if ...
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### Isolating betti numbers from the Euler characteristic of a manifold

I am interested in isolating the Betti numbers of a manifold $\mathcal{M}$ from its Euler characteristic $\chi(\mathcal{M})$ for physical purposes. By isolating, I mean deriving an expression of the ...
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### The suspension of $\mathbb{C}P^2$ is not homotopy equivalent to $S^3 \vee S^5$

I am looking for a relatively simple way to see that $\Sigma \mathbb{C}P^2$ is not homotopy equivalent to $S^3 \vee S^5$. Both the Euler characteristic of a space and its homology groups are invariant ...
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### Gysin -/ Shriek map of projectivized bundle

Consider the Euler sequence $$0\rightarrow \gamma \rightarrow \epsilon^4 \rightarrow Q \rightarrow 0$$ over $\mathbb{P}^3$. Take the projectivized bundle $\pi:P(Q)\rightarrow \mathbb{P}^3$ and ...
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