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.

A chain complex $(A_{\bullet}, d_{\bullet})$ is a sequence $(A_n)_{-\infty}^{\infty}$ of abelian groups (or modules) and group (module) homomorphisms $d_n : A_n \to A_{n-1}$ such that $d_{n-1}\circ d_n = 0$. This data can be represented as follows:

$$\cdots \xrightarrow{d_{n+1}} A_n \xrightarrow{d_n} A_{n-1} \xrightarrow{d_{n-1}} \cdots$$

The homology of a chain complex is the sequence of abelian groups

$$H_n = \frac{\ker d_n}{\operatorname{im}d_{n+1}}.$$

Dually, a cochain complex is a sequence $(A_{\bullet}, d_{\bullet})$ of abelian groups where $d_n : A_n \to A_{n+1}$.

There are many common types of (co)homology including simplicial (co)homology, singular (co)homology, and group (co)homology. A more extensive list can be found here.

Simplicial homology and singular homology are examples of homology theories attached to a topological space. The Eilenberg-Steenrod axioms are a collection of properties that such homology theories share.

For the more abstract aspects of (co)homology theory, the tag may be more appropriate.

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$H_{dR}^k(M/G)\to H_{dR}^k(M)$ is injective

Let $M$ be a smooth manifold and $G$ a finite group of automorphisms acting properly on it. If $\pi:M\to M/G$ is the projection map, prove that $\pi^*:H_{dR}^k(M/G)\to H_{dR}^k(M)$ is injective. I know how to prove that $\pi^*:\Omega^k…
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Show that $\zeta(2,2) = \frac{3}{4} \zeta(4)$ by integrating $\int \frac{dx}{x} \wedge \frac{dx}{x} \wedge \frac{dx}{1-x} \wedge \frac{dx}{1-x}$

On Wikipedia there's an OK discussion of multiple zeta values (MZV). We have an identity: $$ \zeta(2,2) = \sum_{m > n > 0} \frac{1}{m^2 \, n^2} = \frac{3}{4} \sum_{n > 0} \frac{1}{ n^4} = \frac{3}{4} \zeta(4) $$ This formula might already be on…
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Existence of proper maps for compactly-supported Mayer-Vietoris

I was given the following exercise: "Let $\{U,V\}$ be an open cover of a manifold $M$. Prove that there is an exact sequence $$0\to \Omega_c^*(M)\to \Omega_c^*(U)\oplus\Omega_c^*(V)\to\Omega_c^*(U\cap V)\to 0."$$ Bott and Tu say that this can be…
Alex
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Information about the total complex from the second page of a spectral sequence

This is an exercise from Ravi Vakil's notes on spectral sequences. Suppose you have a spectral sequence $E^{\bullet\,\bullet}_\bullet$ such that $E^{i\,j}_0$ is zero if either $i$ or $j$ is negative (so if you let your arrows point upward and…
Mike Pierce
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Cohomology of compact complex nilmanifolds

Given a simply connected nilpotent real Lie group $G$ and a discrete subgroup $\Gamma$ such that the nilmanifold $G/\Gamma$ is a compact complex manifold. Is there any way to calculate (de Rahm) cohomologies of $G/\Gamma$? By Nomizu theorem: …
Ronald
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Fundamental class of diagonal of $\mathbb{R}P^n \times \mathbb{R}P^n$ with coefficients $\mathbb{Z} / 2$

I am having trouble to understand the fundamental class of the diagonal $\Delta \subset \mathbb{R}P^n \times \mathbb{R}P^n$ with coefficient $\mathbb{Z} / 2$ as in the class I only gets a rough idea how to do this using the cohomology mod torsion.…
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Reducing spanning vector elements to {1, 0, -1}

Background I'm working on a project that involves computing the homology of simplicial complexes. The way this is done results in the spanning set of vectors that describes the set of homology groups or holes in the complex, and these vectors are…
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Equivalence between different formulations of homology axioms

For a reduced (co-)homology theory defined for the CW-complexes, is the following formulation equivalent to the axioms of excision and long exact sequence? It states that for a CW-pair $(X,A)$ there are boundary homomorphisms $\partial : h_n(X/A)…
Bargabbiati
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$\mathbb{CP}^k$ not retract of $\mathbb{CP}^n$ when $k < n$

I'm currently reading Massey's book on algebraic topology. I do not have a clear understanding on cohomology ring and how it can be applied to prove the claim that $\mathbb{CP}^k$ is not a retraction of $\mathbb{CP}^n$ when $k < n$. Similarly, in…
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Cup products in compactly supported and twisted cohomology of manifold.

I know very well about the cup product in ordinary cohomology. My question is that how we compute the different cup product with twisted or compactly supported cohomology i.e, $$\cup:H^{*}(M,\mathbb{Q}^{w})\times…
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For path connected manifolds, the orientation double cover is path connected iff manifold is not orientable

The "orientation double cover" $\tilde{M}$ of a n-manifold $M$ is defined as follows: $\tilde{M} = \{\mu_x | x \in M, \mu_x \textrm{ is a generator of } H_n(M,M-\{x\})\}$. Well, if we assume that the n-manifold $M$ is path connected, apparently it…
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Does tensor product commute with homology?

My question is if homology commutes with taking tensor product. I believe in general it is not true but when the tensor product is with a projective module it is. I would like to take a look at a proof but I havent found one yet. I was trying to…
allizdog
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Cohomology ring

There exists a construction of the cohomology ring using only the Eilenberg–Steenrod axioms? I'm not able to find a reference where the theory is developed only with the axioms (I mean all of then, not generalized cohomology).
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Questions regarding Bockstein Spectral Sequence (McCleary's book)

I have some questions regarding Bockstein homomorphism in John McCleary's book (pg 455-456). Q1) Is there a typo, is it supposed to be $\bar{u}\in H_n(X;\mathbb{Z}/r\mathbb{Z})$? Q2) How do we see that $\partial(c)\neq 0$? Q3) How do we conclude…
yoyostein
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Why is the cohomology ring of a H-space a Hopf algebra?

Let $X$ be a finite-type H-space. This means $H^*(X)$ is finitely generated, and there exists a 'multiplication' $\mu:X \times X \to X$ and an 'identity' $e\in X$ such that the maps $x \mapsto \mu(x,e)$ and $x \mapsto \mu(e,x)$ are homotopic to…
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