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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A problem on finding cap product structure on $\mathbb {RP}^{2}$ and Klein's bottle .

$\mathbf {The \ Problem \ is}:$ Compute all the cap products for $\mathbb{RP}^{2}$ with $\mathbb{Z}$ and $\mathbb{Z} / 2$ coefficients. Do the same for the Klein's bottle $K$. $\mathbf {My \ approach}...
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Intuition between the equivalence between Cech and Singular Cohomology?

We know that, under suitable assumptions, the Cech Cohomology of a topological space is isomorphic to the singular cohomology. The proof seems to be mostly algebraic. I am wondering: is there a ...
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Why are (co)homology groups algebraic invariants?

Background & definitions: I am studying algebraic topology, in particular (co)homology groups and Mayer-Vietoris sequences. Important terms: Algebraic invariant = property of a topo. space ...
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What do the elements of the chains of a simplicial complex represent?

I've just started to learn homology and I don't quite understand why we define chains the way we do. For a simplicial complex $S$ we define $C_k$ to be the $k$-chains on $S$ given by an abelian group ...
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Equivalent condition for $\mathrm{Tor}_{n+1}^R(G,N)=0$

I am reading Relative Homological Algebra by Enochs. In the proof of lemma 9.1.4, he said '$0 \to S \otimes N \to P_n \otimes N$ is exact since $\mathrm{Tor}_{n+1}(G,N)=0$'. Besides, he said '$0 \to S ...
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Combinatorial Laplacian for homology with $Z_2$ coefficients

Consider I have boundary operators $\partial_1$, $\partial_2$: $\partial_1 \circ \partial_2 = 0$. Then if interested in $\text{ker}\,\partial_1 / \text{im}\,\partial_2$ one can study $\text{ker}\,(\...
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3 votes
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Computing singular homology groups of quotient space

I want to compute the homology groups of $X$, the quotient of $S^2 \times S^1$ by the relation $(x,z) \sim (-x,-z)$. I've already computed the homology groups of $S^2 \times S^1$ using Mayer-Vietoris (...
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Cohomology of matrix groups acting on vector spaces under change of basis

Let $F$ be a field and let $F^n$ be the vector space of finite dimension $n$ over $F$. Let $G\leq \textrm{GL}_n(F)$. Apply a change of basis to $F^n$, represented by the matrix $C$, and get the group $...
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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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First homology group of a closed non-orientable 2-manifold vía the cellular homology groups

Let $N_h$ be a closed non-orientable 2-manifold of genus $h\geq 1$. I am trying to compute the first homology groups $H_1(N_h)$. For do so, it is sufficient compute the cellular homology group $H_1^{...
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The subcategory ModR is finitely generated R-modules has always cokernels but has kernels only if R is Noetherian

I have to prove that the subcategory modR of finitely generated R-modules has always cokernels but has kernels only when R is a Noetherian ring. I know how to prove that ModR has cokernels is it the ...
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Homology homomorphism of the inclusion map of a path component

I'm quite new learning singular homology and from problem 9.9 from Greenberg & Harper's book of algebraic topology I ran into this problem: Let $X$ to be a path component of $X'$ and let $\iota:X\...
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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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The isomorphism $H^1(X;\mathbb Z_2) \rightarrow \operatorname{Hom}(\pi_1(X),\mathbb Z_2)$ and $w_1(E)$

On page $87$ of Hatcher's book Vector Bundles and K-Theory it states that, assuming $X$ is homotopy equivalent to a CW complex ($X$ is connected), there are isomorphisms $$H^1(X;\mathbb Z_2) \...
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Factorization of quasi-isomorphism is also a quasi-isomorphism

Let $\mathcal{A}$ be an abelian category, $ X_\bullet \overset{f}{\hookrightarrow} Y_\bullet \overset{g}{\hookrightarrow} Z_\bullet$ be in $Ch(\mathcal{A})$ such that $gf:X_\bullet \hookrightarrow Z_\...
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Proving that $S^1 \vee S^1 \to S^1$ induces a projection on homology groups

I am trying to compute the homology of the Klein bottle using the Eilenberg-Steenrod axioms and after a number of steps, I reached a part where I need to show that the map $f:S^1 \vee S^1\to S^1$, ...
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2 votes
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Literature Request: Koszul-Tate Resolutions

Good people! So, these papers that have become incredibly relevant to me all keep using these things called Koszul-Tate resolutions, which I must admit was not part of my training. I know about ...
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tensor product of exterior algebras isomorphic to another exterior algebra

In class (also on page 355 of Fomenko's Homotopical Topology), I learned that $$H^*(SU(n);\mathbb{Z})=\bigwedge[x_3,x_5,...,x_{2n-1}]$$ with $deg(x_i)=i$, and that $$H^*(SU(n);\mathbb{Z})\cong H^*(S^3\...
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Persistent betti numbers and birth and death of classes

I'll copy and paste the background information in my other question: Given a filtration of complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$, we have induced homomorphisms $...
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Proof that $\mathbb{RP}^2$ is not the suspension of a space $X$.

I am looking to prove that $\mathbb{RP}^2$ is not homeomorphic to $\Sigma X$ for any space $X$, where $\Sigma$ denotes suspension of a space. I am stumpted as the context of the question is homology, ...
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5 votes
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Order relation between cohomological dimensions of open orientable manifolds

Let $M$ be an open orientable connected manifold and let $\operatorname{Cohdim}_{\mathbb{Z}_{2}}(M)$ and $\operatorname{Cohdim}_{\mathbb{Z}}(M)$ be the cohomological dimensions of $M$ over $\mathbb{Z}...
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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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2 votes
1 answer
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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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Understanding the Poincare Duality Map (with Torus)

I think considering an explict example with computation may help me to understand the Poincare duality. Consider the torus $\mathbb{T}^2$. Let $$D : \mathrm{H}^1(\mathbb{T}^2;\mathbb{Z}) \to \mathrm{H}...
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Compactly supported cohomology of open disk plus a point

Let $X=\{v\in\Bbb R^2:\|v\|<1\}\cup\{(-1,0)\}$, that is, the open unit disk union a point on its boundary. (Here, $(-1,0)$ refers to a point in the plane, not an interval.) What is $H_c^•(X)$, that ...
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1 vote
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Motivation for Hypercohomology

Someone can explain to me this bit from the wikipedia article on hypercohomology: https://en.wikipedia.org/wiki/Hyperhomology It turns out hypercohomology gives techniques for constructing a similar ...
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Why is any map $f:M \to \mathbb{CP}^\infty$ homotopic to a map $f_0:M \to \mathbb{CP}^1$ if $M$ is a 3-manifold?

I am reading a proof of the result that every element of the first homology a closed, oriented 3-manifold $M$ can be represented by a knot in $M$. This seems to be a pretty standard result, but I am ...
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Euler characteristic of odd dimensional manifold - Hatcher

I ran into some trouble while reading through Hatcher's proof of the following: Corollary 3.37. A closed manifold of odd dimension has Euler characteristic zero. There is only one part of the proof ...
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Prove the first Stiefel-Whitney class of the line bundle (as a Mobius strip) over the base space $S^1$ is $w_1=1$

What I know: A trivial vector bundle $E$ has a vanished first Stiefel-Whitney class $w_1(E)=0$. If a vector bundle $E$ is non-orientable (as a vector bundle, not a manifold), $w_1(E)\neq 0$. The ...
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  • 151
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The Koszul complex is invariant

I am studying Koszul complex over a commutative noetherian local ring. I see the following propertie: If $I$ is an ideal, $x=x_1,\dots ,x_n$ and $y=y_1,\dots,y_n$ are minimal system of generators of $...
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Computing singular homology of a cylinder with a bottom, using Mayer Vietoris Sequence

I am trying to practice using MVS on an easy example, a cylinder with a bottom. Explicitly, something like $S^1 \times [0,1]$ with a copy of $D^2$ glued at one end. Call this object $X$. Then I ...
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5 votes
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Applications of the cup product before descending to cohomology

The cup product is a map $H^p \times H^q \to H^{p+q}$ which turns the cohomology of a (nice) space $X$ into a ring (commutative in the graded sense). It can be understood geometrically in many ways (...
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If $E_r^{p,q}\cong {E_r'}^{p,q}$ for $r=\{r_0,\infty\}$ and all $p,q$, then $E_r\cong E_r'$

Let $\{E_r\}$ and $\{E_r'\}$ be two (cohomological) spectral sequences of vector spaces (to avoid extension problems). Suppose that, for certain $r>0$, $$ E_r^{p,q} \cong {E_r'}^{p,q} \qquad \...
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Rational maps and homology

Let $X$ and $Y$ be algebraic varieties defined over $\mathbb{C}$. By considering them equipped with the complex topology, we can talk about singular chains and singular homology, as usual Suppose $f:X\...
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A formula for homology groups of cartesian product of finitely many spheres

By Kunneth Formula, for positive integers $m,n$, if $m\neq n$, then $$H_p (S^m \times S^n)=\begin{cases}\mathbb{Z} & \text{if}\; p=0,m,n,m+n\\ 0, & \text{otherwise}\end{cases}$$ and if $m= n$,...
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Simplicial Homology Groups of Circle Wedge a Torus

Compute the simplicial homology groups of $S^1 \vee (S^1 \times S^1)$ in all dimensions. I'm trying to practice simplicial homology, and want to make sure I understand at a technical level what's ...
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1 vote
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Does the 3-manifold $S^1\times S^2$ bound a smooth integral homology ball?

Does the 3-manifold $S^1\times S^2$ bound a smooth (integral) homology ball? The only 4-manifolds I know whose boundary is $S^1\times S^2$ are $S^1\times D^3$ and $D^2\times S^2$, and both are not ...
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Reduced homology of disjoint union

For non-reduced singular homology we know that the inclusions $\imath_i : X_i \hookrightarrow \bigsqcup_{i \in I} X_i$ to a disjoint union induce an isomorphism $\oplus_{i \in I} (\imath_i)_{\ast} : \...
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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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Compute the $\pi_{1} (S^{3} /G)$ and $H_{m} ( S^{3} / G)$ for the following $G$

I am studying for a qualying exam and there is one exercise from the previous years exams which I don't know how to approach. Let $n$ be a positive integer and $$ G = \{ g \in \Bbb C^{\times} | g^{n} =...
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Homology of the Eilenberg-MacLane spectrum of Fp with coefficients in Fq for p and q prime

I understand the Steenrod algebra for $\mathbb{F}_{p^n}$ both from classical calculations and a past question, but I'd like to ask about $H\mathbb{F}_{p*} H \mathbb{F}_q$ for $p$ and $q$ different ...
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On cohomology ring of a mapping cone .

$\mathbf {The \ Problem \ is}:$ Let $\eta: S^{3} \rightarrow S^{2}$ be the Hopf map, that is, the attaching map for $C P^{2}$. Consider the map $\tau$ expressed as the composite $$ \tau: S^{3} \...
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3 votes
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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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Action of a map on homology

I'm currently studying algebraic topology from Hatcher's text, and I came across the following problem from an old qualifying exam: The coefficient sequence $0 \rightarrow \mathbb{Z} \xrightarrow{p} \...
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