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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.

4
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1answer
38 views

When do elements of $\operatorname{Hom}(G,G)$ correspond to invertible self maps of $K(G,n)$?

Suppose we pick a natural isomorphism between $H^n(-;G)$ and $\langle -, K(G,n)\rangle$, when does an element of $H^n(K(G,n),G)=\operatorname{Hom}(G,G)$ correspond to a self map of $K(G,n)$ that has a ...
0
votes
1answer
25 views

Cohomology of classifying space

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces: 1) $ H^\ast (BSU(2), \mathbb{Z}) $ 2) $ H^\ast (BO(3), \...
0
votes
1answer
10 views

Reference for the cohomology of SU

Let SU be the infinite special group. Where can I find the following fact (state in part III of the Adam's blue book): $H^{6}(SU,Z)=0$. Thank you.
2
votes
0answers
29 views

Wu formula for $\mathbb{Z}_N$ classes

Let $w_1\in H^1(M, \mathbb{Z}_2)$ be the Stiefel-Whitney class of the tangent bundle of d-dimensional manifold $M$, and $x_{d-1}\in H^{d-1}(M, \mathbb{Z}_2)$. Wu formula tells us $$Sq^1 x_{d-1}= u_1\...
1
vote
1answer
31 views

Homology of the complement of a compact subset inside ball

Let $A$ be a abelian group and let $\mathbb{B}^m=\{|x|<1\}$ and let $L\subseteq \mathbb R^m$ be compact such that $K:=L\cap \mathbb B^m\ne\emptyset$ is connected. Clearly we have an iso $$H_{m-1}(\...
0
votes
1answer
24 views

detecting flares with persistent homology

Can persistent homology detect "flares" how does it do so, if it can. I know persistent homology can certainly find "loopy" structure, like noisy circles, but I'm not sure about "flares".
2
votes
0answers
23 views

Quasi-isomorphism of injective complexes is a homotopy equivalence?

Let $A^* = (0\rightarrow A^0\rightarrow A^1 \rightarrow ...), B^* = (0\rightarrow B^0\rightarrow B^1 \rightarrow ...)$ be complexes of injective objects of an abelian category $\mathcal{A}$. Suppose ...
0
votes
0answers
51 views

Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
0
votes
0answers
33 views

Thom class in homology for defining orientability of vector bundles

A rank $r$ real vector bundle $p : E \to B$ is said to be orientable if there is a Thom class $\tau \in H^r (D(E),\partial D(E) ; \mathbb{F})$ (where $D(E)$ is a unit disk bundle and $\mathbb{F}$ a ...
2
votes
1answer
41 views

Serres Vanishing Theorem III

I am trying to understand the Lemma 29.3.1. We begin with a quasi-compact scheme. Then we have the following construction: Let $X$ be a scheme. $x \in X$ a closed point, $U=Spec(A)\subseteq X$ ...
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0answers
16 views

Inhomogeneous Lie derivative equation on a Lie group

Let $G$ be a connected Lie group and let $\xi_i$, $i=1,...,n$ be a basis of its Lie algebra (say, of left-invariant vector fields). We let $B_i$, $i=1,...,n$ be given symmetric sections of $TG \otimes ...
0
votes
1answer
15 views

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 ...
2
votes
1answer
64 views

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}...
1
vote
1answer
66 views

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 $...
2
votes
1answer
42 views

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 ...
2
votes
1answer
22 views

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. ...
1
vote
1answer
46 views

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 $\...
2
votes
1answer
61 views

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 ...
0
votes
1answer
41 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\...
1
vote
0answers
52 views

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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0answers
28 views

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 ...
4
votes
0answers
40 views

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 ...
2
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0answers
38 views

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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vote
0answers
42 views

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 ...
6
votes
0answers
80 views

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(...
2
votes
1answer
20 views

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 ...
2
votes
0answers
44 views

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 ...
0
votes
0answers
14 views

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+...
2
votes
1answer
33 views

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: $\...
1
vote
0answers
17 views

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 ...
1
vote
1answer
64 views

Proof of Kunneth theorem [on hold]

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 ...
1
vote
0answers
41 views

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 ...
0
votes
1answer
37 views

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 $\partial S$ ...
4
votes
1answer
54 views

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, ...
6
votes
0answers
49 views

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 ...
5
votes
2answers
57 views

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 ...
7
votes
1answer
87 views

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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vote
0answers
33 views

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 ...
0
votes
0answers
14 views

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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votes
0answers
23 views

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 $(H_{dR}(M),\wedge)$....
2
votes
1answer
38 views

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 ...
2
votes
1answer
30 views

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}...
8
votes
0answers
93 views

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 ...
2
votes
0answers
30 views

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 ...
0
votes
0answers
15 views

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?
0
votes
0answers
10 views

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 ...
1
vote
1answer
36 views

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 ...
2
votes
0answers
43 views

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.
5
votes
0answers
55 views

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 ...
2
votes
1answer
42 views

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 ...