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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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Tensor product of a (co)chain and a cochain complexes

This is a question about the grading convention. Suppose we have a chain complex $$C_\bullet = 0\to C_n\stackrel{d_n}\to C_{n-1}\stackrel{d_{n-1}}\to...\to C_1\stackrel{d_1}\to C_0\to 0$$ and two ...
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1answer
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Definition of Prism Operator and homotopy of chain complexes

I am reading Algebraic Topology by Allen Hatcher. For reference: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf On page 112, in the first sentence of the last paragraph, we find the following: The ...
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1answer
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Homology Represents topological Subspaces

Consider $X:= \mathbb{PC}^n $ the projective space. It is well known that the integral homology of $X$ is $H_i(X, \mathbb{Z}) = \mathbb{Z}$ is given by: $0 \leq i \leq 2n$ even, and $H_i(X, \mathbb{Z}...
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1answer
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Show that $\bigoplus_{i \text{ even}}C_i=\bigoplus_{i\text{ odd}}C_i$

Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is sufficiently large and that for all $...
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1answer
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Universal covering that induces zero on homologies

Let $p:\tilde{X}\rightarrow X$ be the universal covering space such that $p_*$ is zero on all homologies of dimension greater than zero. Does this imply that $X$ is $K(\pi_1(X),1)$? Working with the ...
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1answer
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Computing the homology of a simple chain complex

Let $R$ be a ring and $x\in R$ be a central element. Consider the complex $$0 \rightarrow R \xrightarrow{x} R \rightarrow 0$$ concentrated in degrees 1 and 0. Compute the homology of this complex. ...
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1answer
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Map of H-spaces inducing zero on homologies

If a map of $H$-spaces $f:X\rightarrow Y$ induces zero on the homology groups at dimensions greater than zero does it necessarily induce zero map on the homotopy groups? It is definitely true for $\...
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1answer
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Path Connectedness of Simply Connected Space Minus a Point

Suppose that $X$ is a simply connected topological manifold of dimension at-least $2$. Fix a point $x \in X$ and define $\tilde{X}\triangleq X-\{x\}$. How can I prove that the $0^{th}$ signular ...
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1answer
37 views

Serre's Vanishing Theorem

I am having problem with the construction in Serre's Vanishing Theorem. The proof begins with a general construction which I don't follow. Let $X$ be a scheme. $x \in X$ a closed point, $U\...
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Role of d^2 = 0 in chain complex

What is the motivation for requiring that the square of a differential be 0 for a complex, aside from enabling us to speak of the homology of a complex? Other homological notions like chain maps, ...
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isomorphism for real vector bundles and complex vector bundles with inner product.

Let $B$ be paracompact. Call $\mathrm{Vect}_{\mathbb R}(B)$ the category of real vector bundles over $B$ and $\mathrm{Vect}_{\mathbb C,f}(B)$ be the category of complex vector bundles with an inner ...
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Homology of the maps between complex projective spaces

Suppose $m > n$, and let $f : \mathbb CP^m \to \mathbb CP^n $ be continuous, the claim is that the induced map between the homology(over $\mathbb Z$) is zero. I have no clue why this should be ...
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Why does this exact sequence exist?

I'm reading a proof and don't understand a certain part. Let $A^\bullet$ be a (cochain) complex of abelian groups. Let $I^\bullet$ be an injective resolution of an abelian group $B$. Then there is a ...
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Is this an example of a local system

In Bott, Tu, one is asked to calculate the cohomology of the following sheaf: Exercise 10.7 (Cohomology with twisted coefficients). Let $\mathscr{F}$ be the presheaf on $S^1$ which associates to ...
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1answer
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Integral homology group of a 3-torus cut out a donut

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...
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1answer
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If $A$ is a weak deformation retraction of $X$, is $i*$ an homology isomorphism?

Let $A \subseteq X$ be a weak deformation retraction as in this definition: https://topospaces.subwiki.org/wiki/Weak_deformation_retraction Does this mean that $i* : H_n(A) \to H_n(X)$ is an ...
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Long exact sequence of cohomology

I was reading about cohomology and long exact sequences. I found that Given $$0 \to L \to M \to N \to 0$$ is a short exact sequence of $G$- modules, then a there exists a long exact sequence is ...
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If $X$ is contractible and $A \subseteq X$ then $H_{n}(X,A) \approx \tilde{H}_{n-1}(A)$

If $X$ is contractible and $A \subseteq X$ then $H_{n}(X,A) \approx \tilde{H}_{n-1}(A)$ I'm a little stuck with this exercise. What I've got so far is just the LES: $$ \dots \to H_{n+1}(X) \to H_{n+...
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1answer
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If $A$ is a weak deformation retraction of $X$ then the relative homology $H_n(X,A)$ is trivial for all $n$

I want to prove that if $A$ is a weak deformation retraction of $X$, then the relative homology $H_n(X,A)$ is trivial for all $n$. I would like to prove this by induction. Studying the LES, I get: $\...
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About deformation retract of pairs

Here: Relationship between homology of suspension of $X$ and $X$, Joe Moeller argued that $(\Sigma X\setminus U,C_-^n\setminus U)$ deformation retracts to $(C_+^n,X)$. I don't know the definition of ...
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1answer
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Proof of Kunneth theorem

What are different ways to prove Kunneth theorem relating singular homology of product space $X * Y$ in terms of homology of $X$ and $Y$? or reference?I know some ways: use cell homology for cell ...
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Homology group of the total space and base space of a vector bundle

Let $\pi: E\to B$ be a vector bundle with fibre $F$ where $F$ is a vector space over $\mathbb{R}$, is the homology group $H_n(E)=H_n(B)$? I suppose that $E$ and $B$ are homotopic. I can see that ...
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1answer
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Homology group of a triangulation(simplicial complex) space

Let $X$ be a connected oriented triangulation(polyhedron) space, i.e., homeomorphic to a geometric realization of an oriented simplicial complex $S$ with dimension $n$, and the boundary $\partial S$ ...
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1answer
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Making $H^*(\mathbf{P}^\infty)=\lim H^*(\mathbf{P}^n)=k[t]$ precise using stacks

The stack $B\mathbf{G}_m$, i.e. morally $\mathbf{P}^\infty$, has (etale) cohomology $\mathbf{Q}_\ell[t]$. The scheme $\mathbf{P}^n$ has cohomology $\mathbf{Q}_\ell[t]/t^n$. In algebraic topology, ...
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Fixed-point free action of $\mathbb{Z}/p\mathbb{Z}$ on a finite CW complex

Problem 2a on this old qualifying exam asks the following: Suppose $X$ is a finite CW complex and $X$ admits a fixed-point free action of $G:= \mathbb{Z}/p\mathbb{Z}$ for some prime $p$. Prove ...
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2answers
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The cohomology ring of an oriented closed $3$-manifold with $\pi_1=\mathbb Z$

Let $M$ be a closed, orientable, connected manifold of dimension $3$, such that $\pi_1(M)=\mathbb Z$. Find its cohomology ring $H^*(M;\mathbb Z)$. Clearly $H^0=H^3=\mathbb Z$. Now since connected ...
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1answer
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If every map $X\to S^1$ is nullhomotopic, then $H_1 (X)$ is finite, without using cohomology?

If $X$ is a finite connected CW complex, then $H_1 (X, \mathbb{Z})$ is finite iff every map $X \to S^1$ is nullhomotopic. This question has already been asked and answered here. However, the problem ...
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Cohomology with compact supports of infinite trivalent tree

I want to find the cohomology with compact supports over $\mathbb Z$ of a) Figure 1, consisting of three rays emerging from a point, and b) Figure 3, the infinite trivalent tree. Since both are ...
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Cup product of the same cochains

A.Hatcher's p207 See the picture in A. Hatcher's p207. He says that the cup product $\alpha_1\cup\alpha_1$ is zero. But I think that I should get a -1 on the red triangle if the vertex order is ...
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Exterior Product on de rham homology

Given a smooth manifold $M$ and its differential graded commutstive de Rham algebra $(\Omega(M),d,\wedge)$, the wedge product $\wedge$ can be projected onto the de Rham cohomology $(H_{dR}(M),\wedge)$....
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1answer
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Effect of the multiplication map $G\times G\rightarrow G$ on homology groups

Let $G$ be a topological group with multiplication $m:G\times G\rightarrow G$. Let $\omega$ denote the composition $G\vee G \subset G\times G\xrightarrow{\text{m}}G$ I have to calculate the effect of ...
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1answer
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Why is $Tor_0(A,B) = A\otimes B$?

We define a free resolution of an abelian group $A$ to be a level-wise free chain complex $\tilde A$ such that $H_0 (\tilde A) = A$. Also, we define $Tor_k (A,B)$ to be $H_k(...\rightarrow \tilde{A_2}...
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Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
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Homology of solvable Lie algebras

Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
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Cup product without cellular decomposition?

As we know, the cup product does not rely on the particular cellular decomposion of the space. So, is it possible to define the concept of cup product not rely on the cellular decomposition?
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About Lie algebra cohomology and Ext group

Let $\mathfrak{g}$ be a Lie algebra over a field $K$. Then the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $V$ is the right derived functor of $V\mapsto V^\mathfrak{g}$ and can be ...
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1answer
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trivial modules of group rings

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any simple $R$-module. If one knows that $H^0(D,M)=0$, is $M=0$? If not, under what further conditions can one ...
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concerning about homology of a graph

I'm very lost in the following Bredon problem 9-pag 240 Let $G \subset S^n$ be a finite connected graph. Find $H_{\bullet}(S^n \setminus G)$. Any hint would be appreciated. Thanks.
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Cohomology of quadric in $\mathbb{C}^4$ at infinity

How to prove that the singular cohomology of $X=V(xy-zw)\setminus 0 \subset \mathbb{C}^4$ is $$H^*(X,\mathbb{Z})=\{\mathbb{Z},0,\mathbb{Z},\mathbb{Z},0,\mathbb{Z}\}?$$ Preferably, I would like to know ...
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1answer
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Evaluating a symplectic form on $\pi_2$ or its image through the Hurewicz map

Let $(M,\omega)$ be a symplectic manifold. There are a priori two ways of evaluating $\omega$ on an element $A \in \pi_2(M)$: we can integrate $\omega$ on any representative $u : S^2 \to M$ of the ...
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1answer
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Group homology of action on left cosets

If $G$ is a group and $H$ a subgroup which is not normal. What is the homologies of the action of $G$ on the left coset space $G/H$? The action is multiplication from left. More precisely, $G/H$ is ...
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The cohomology ring of two tori glued along the first circle

Let $X$ be the space consisting of two tori glued along the first circle: I interpret the space as $X=S^1_{(1)}\times S^1_{(2)}\sqcup S^1_{(3)}\times S^1_{(4)} / (S^1_{(1)}\sim S^1_{(3)})$. I want ...
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1answer
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Relationship between Betti number and Genus

I recently found out that Mathworld gave the same definition for both Betti number and Genus as: "the largest number of nonintersecting simple closed curves that can be drawn on the surface without ...
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1answer
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How is the generator of the first homology of the torus non-trivial?

Consider the above representation of the torus $X$. I need to show that if $\phi\in C^1(X;\mathbb Z)$ is the cochain that takes the value $1$ on the red lines with the orientation given by the ...
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Serre fibration and Mittag Leffler condition

The proof I am concerned is 2.2.5 pg 37 Kochman Stable Homotopy. Let $R$ be a commutative ring. Let $S^n \rightarrow E \xrightarrow{p} B$ be Serre fibration. $B$ a CW complex, $B$ simpliy ...
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1answer
57 views

$\mathbf{R}P^{n}/ \mathbf{R}P^{n-2} \simeq S^n \vee S^{n-1}$ iff $n$ is odd

Prove that $\mathbf{R}P^{n}/ \mathbf{R}P^{n-2} \simeq S^n \vee S^{n-1}$ iff $n$ is odd. One side is solved just calculating the homology groups of both sides. I can't see if $n$ is odd how to show ...
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1answer
31 views

Reference for Universal Coefficient Theorem

I am looking for a proof of the following fact: let $C$ be a chain complex of real vector spaces, $C^*$ the dual cochain complex. Then $H^n(C^*) \cong H_n(C)^*$. That is, taking homology commutes with ...
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1answer
88 views

Computing algebraic de Rham cohomology

Let $R=\mathbb C[x,y]/(y(x-a)(x-b)-1)$ where $a,b$ are distinct complex numbers. Show that the cohomology of the de Rham complex $$0\to R\to \Omega_{R/\mathbb C}\to 0$$ is $\mathbb C$ in degree zero ...
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1answer
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Understanding the proof of a criterion of equivalent group extensions

I stuck in a step of the proof of the equivalence of two group extensions for 30 mins. In the place of red arrow, how does the previous line deduce the next? PS: The author denote the operation in $G,...
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1answer
53 views

Show that there exists a closed, orientable manifold for any Euler characteristic

I'm trying to show that there exists a closed, orientable manifold $M^n$ with $\chi(M^n)=k$ for every $k\in\mathbb{Z}$. I was able to show that such a manifold has even dimension using Poincare ...