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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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Wang Sequence for the circle $S^1$

Let $F\stackrel i \to E\stackrel \pi\to S^1$ a fiber bundle over the circle $S^1$. There is a long exact sequence sequence in cohomology, called Wang: $$\dots\to H^k(E)\stackrel {i^*}\to H^k(F)\...
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1answer
49 views

Computing cohomology of dihedral group in detail

So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, \Bbb Z)$. using Lyndon SS. I have obtained a few obstacles: My computation, using the fact that there is a $C_2$ action ...
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0answers
8 views

“Restriction” map in group homology, what was meant? Rotman

Def 1: $\alpha:G \rightarrow G'$, group homomorphism. If $A'$ is a $G'$ module $f:A \rightarrow A'$ is a $\Bbb Z$ map we call $(\alpha, f)$ a compatible pair if $f:A \rightarrow _{\alpha}A'$ is a $G$-...
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1answer
43 views

Base change morphisms for higher pushforwards

Consider a Cartesian diagram consisting of morphisms of schemes. StackExchange doesn't support TikzCD, but the following hopefully suffices. $$\begin{array}{ll} X' & \stackrel{\psi'}{\rightarrow}...
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vote
1answer
47 views

Cellular homology of 3 Torus (Clarification) , Hatcher

This is on pg 143 and is also asked here which I quote: We given $T^3$ the $3$-torus a cell decomposition as follows: $1$ $3$-cell , $3$ $2$-cell, $3$ $1$-cell and $1$ $0$-cell. Giving cellular ...
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0answers
27 views

Any chain map of Moore complex null homotopic? Can't find mistake

Suppose that $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ are cosimplicial abelian groups. Let $f, g: A\to B$ be two maps of cosimplicial abelian groups. Let $s(A)$ and $s(B)$ denote the Moore (co)chain ...
5
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1answer
45 views

Computing Pontryagin Square

Suppose $v$ is a $\mathbb{Z}_2$ cochain on a four dimensional spin manifold $M$, i.e. $v\in H^1(M, \mathbb{Z}_2)$. I am interested in evaluating the quantity $$\exp \bigg(i \frac{\pi}{2}\int_M \...
3
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1answer
23 views

Induced map on cohomology being zero impies null-homotopic?

Let $f:A^\bullet\to B^\bullet$ be a morphism of chain complexes (of any give abelian category). We know that if $f$ is homotopic to the zero map, then $f$ will induce zero map on cohomology. I want to ...
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0answers
36 views

Axioms for the generalized cohomology

Here I would like to understand in the homotopy axiom what is the induced homomorphism on (co)homology? What it means and how is the induced homomorphism defined?
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1answer
52 views

Showing that the sum of any 1-cycle and its opposite orientation is a 1-boundary

Suppose that $\phi$ is a $1-cycle$ in a topological space $X$, and that $\phi^*:= \phi r$ is its opposite orientation where $r(t,1-t)=r(1-t,t)$. How should one show that there is a $2-chain$ whose ...
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1answer
34 views

A question about universal coefficient theorem

Assume that $M$ is $n$-dimensional, compact, connected and oriented with boundary manifold. Show that if $H_{k-1}(M,\partial M, Z)$ is torsion-free, then $H_{n-k}(M, Z)$ is torsion-free. This ...
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0answers
24 views

Singular cohomology with compact support of $X \times \mathbb{R}$

Let $X$ be a topological space. I would like to show that $H^n_c(X,G) \cong H^{n+1}_c(X \times \mathbb{R},G)$ for every abelian group $G$ and for every $n \in \mathbb{N}$. This is an exercise from ...
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1answer
22 views

is it true that surjective chain map induce surjective homomorphism between homology groups?

Need surjective chain map induce surjective homomorphism between homology groups? Since $f_*[x]\to[f(x)]$?
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0answers
43 views

Cohomology groups $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$

I'm trying to compute $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$. Here the semi-direct product is given by the outer automorphism of $PSU(4)$. By Serre spectral sequence, we have $...
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0answers
30 views

$\widetilde{H}_n(CX,X)\cong \widetilde{H}_n(SX) $

I want to relate the homology of the suspension and the cone of a space by proving they are equal: $\widetilde{H}_n(CX,X)\cong \widetilde{H}_n(SX) $. Here $ CX$ is the cylinder with all the base ...
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0answers
39 views

understanding Thom's theorem for $\mathrm{MU}$ and Milnor-Novikov's result.

Sorry if this has been asked before. $\newcommand{\MU}{\mathrm{MU}}$ I'm trying to understand the complex cobordism spectrum $\MU$, but I don't fully understand Thom's theorem, namely, that $$ \MU_\...
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1answer
23 views

Showing that interior of a cover of a topological space being a cover is essential to guarantee an isomorphism between homology groups

Suppose that $U$ is a cover of a topological space $X$ such that its interior too is a cover. So there is an isomorphism between the homology of $S_n^U(X)$ and that of $S_n(X)$. In Vick’s algebraic ...
3
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3answers
49 views

Wedge sum of spheres is the quotient $X^n/X^{n-1}$

As in the title, I want to prove that $\bigvee_jS_j^n=X^n/X^{n-1};\ X$ is a $CW$ complex and $X^n$ and $X^{n-1}$ are the $n-$ and $n-1$-skeleta. Below, I present a sketch of an attempt using pushouts, ...
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0answers
23 views

Preserve kernel and left-exact

This question has been answered here here. But I still have an issue about it. What is the definition of "preserve kernel" at all. Is it $F(\ker(f))=\ker(F(f))$ or just isomorphic? $$0\to FA\to FB\to ...
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0answers
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Cohomology of $(S^2\times S^2)/\mathbb{Z}_4$

There was a similar question. Let $X=(S^2\times S^2)/\mathbb{Z}_4$ where $\mathbb{Z}_4$ acts on $S^2\times S^2$ as $(x,y)\mapsto(-y,x)$. My question: What are the cohomology rings of $X$ with ...
3
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0answers
47 views

Calculating cohomology of sheaves

I am trying to prove that the twisted cubic $C: (u,v)\rightarrow(u^3,u^2v,uv^2,v^3)$ has as resolution $$ 0\rightarrow \mathcal O(-1)^2\rightarrow \mathcal O^3\rightarrow \mathcal I_C(2)\rightarrow0, $...
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0answers
33 views

degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
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0answers
24 views

When does the Chern character of a line bundle live in the rational cohomology ring?

I just read the definition of the Chern character of a line bundle $L\to B$: $$ch(L) := \sum_{m=0}^{\infty}\frac{c_1(L)^m}{m!}$$ So actually $ch(L)$ lives in $\prod_{i=0}^{\infty} H^i(B;\mathbb{Q})$ ...
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0answers
22 views

Is there a converse to “knowing a good cover” → “knowing cohomology”?

When looking at smooth manifolds, knowing a “nice” cover of our space enables us to calculate the De Rham cohomology via e.g. the Meyer-Vietoris sequence. For instance, $$ M := \{(x,y)\mid x^2+y^2\...
2
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1answer
47 views

$\sum_{i=0}^{n} (-1)^i l(H_i)=\sum_{i=0}^{n} (-1)^i l(G_i)$ [duplicate]

Let $$0 \overset{d_{n+1}}{\rightarrow} G_n \overset{d_n}{\rightarrow} G_{n-1}\overset{d_{n-1}}{\rightarrow}\ldots\overset{d_2}{\rightarrow}G_1 \overset{d_1}{\rightarrow} 0$$ be a sequence of ...
1
vote
1answer
45 views

Homology and embedded surfaces of genus $g$

Suppose to have a closed manifold $M$ of dimension $n\geq 3$ and suppose that there is an element in Homology $[S]\in H_2(M,\mathbb{Z})$ that is represented by a surface $S$ homeomorphic to the 2-...
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1answer
41 views

why the boundary of loop is zero?

In simplices complex, it is known that boundary of boundary is zero $$\partial_{n-1} \partial_{n} \sigma_{0,\ldots,n} = 0$$ where $\sigma_{0,\ldots,n}$ is a n-simplice consists of $n+1$ point, which ...
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1answer
29 views

How to determine the map between the singular homology groups of a torus induced by a linear map $f:\mathbb R^2 \to \mathbb R^2$?

Let $f:\mathbb R^2 \to \mathbb R^2$ be the linear map whose matrix under the standard basis is $ \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix} $. Clearly $f$ induces the map from $\mathbb R^2/...
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0answers
29 views

Alexander duality on (co)chain level

In the book by Stöcker & Zieschang, the Poincaré duality is obtained by an isomorphism $\gamma \colon \operatorname{Hom}(C_q(K), \mathbb{Z}) \to C_{n-q}^{\ast}(K^{\prime})$, $\gamma(\varphi) = \...
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1answer
26 views

In/surjective maps induce mono/epimorphisms in homology?

Recently I came across a proposition: Proposition. The projection $\pi:D^n\rightarrow D^n/S^{n-1}$ induces isomorphism of relative homology $\pi_*:H_p(D^n,S^{n-1})\rightarrow H_p(D^n/S^{n-1},pt)$. ...
0
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1answer
54 views

Realizing a CW Complex as an Adjunction Space: Munkres' Proof

Suppose $Y$ is a $CW$ complex, of dimension $p-1,\ \sum B_{\alpha}$ is a topological sum of closed $p-$ balls. Then, if $g:\sum \partial B_{\alpha}\to Y$ is a continuous map, the adjunction space $X=Y\...
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0answers
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Classifying the spaces that Eilenberg-Steenrod axioms determine the cohomology of

We know that for sufficiently nice spaces, (e.g. spaces with the homotopy type of a CW complex) the Eilenberg-Steenrod axioms determine the ordinary cohomology of the space. One can construct nasty ...
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0answers
11 views

Constructing an injective resolution for a bounded below cochain complex

Let $\mathcal{A}$ be an abelian category with enough injectives. If $X^{\bullet}$ is a bounded below complex, it is a well known fact that you can obtain a bounded below complex $I^{\bullet}$ of ...
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0answers
25 views

Cohomology and left invariant 1-forms

I'm computing the de Rham cohomology of the group $SU(2)$, with $n_g$ generators, making use of the base of left invariant 1-forms $\eta^i, i = \{1, ..., n_g\}$, in order to apply the following ...
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0answers
40 views

$H^*(\mathbb{G}(k,n))$ free with basis the partitions that index the Schubert varieties [duplicate]

I have to proof the fact that $H^*(\mathbb{G}(k,n))$ is, as an abelian group, free with basis the partitions that index the Schubert varieties, but I'm having trouble doing it myself. Denote $\sigma_{...
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0answers
26 views

Does $H^{\bullet}(G, \mathbb{Z})$ have a coalgebra structure?

Here are two well-known facts: Let $X$ be a topological space. We always have the diagonal map $\Delta :X\to X\times X$ and this induces a map $H^{\bullet}(X)\otimes H^{\bullet}(X) \simeq H^{\bullet}(...
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1answer
48 views

Cohomology in groups

I'm trying to find the Rham cohomology of the groups $SU(2)$ and $U(2)$. I know that $SU(2)$ is isomorphic to $S^3$ but I don't know what is $U(2)$ isomorphic to. My question is: if $SU(2) \simeq S^3$ ...
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0answers
49 views

Homology Groups of Oriented Simply Connected Manifold

Let $M$ be a , connected, compact, $\mathbb{Z}$-orientable topological $n$-manifold. The latter $\mathbb{Z}$-orientable means that $H_n(M; \mathbb{Z})=\mathbb{Z} $ holds. Let futhermore $M$ be ...
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2answers
54 views

The $n$th homology of convex subsets of Euclidean space using homotopic maps

I confronted a problem when I was reading the theorem stating that if $f_0$ and $f_1$ are homotopic maps (not to be confused with chain homotopy notion) from $X$ to $Y$, then they induce the same ...
4
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1answer
85 views

Is there any Kunneth formula for homology group with cofficient in an abelian group

I am reading Hatcher Chapter V on spectral sequence. This is a paragraph after Theorem 5.3: The Kunneth formula and the universal coefficient theorem then combine to give an isomorphism $$H_n(B\...
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0answers
25 views

Software for calculating simplicial homology (over the integers)

I am interested in looking for a software package that can calculate the homology of simplicial complexes over the integers. (Ideally, written in a "popular" language such as Python, Matlab, etc. I ...
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votes
1answer
37 views

Proof verification: the homology groups of the cube.

I was so proud of this solution about finding the homology groups of the cube $I\times I \times I$ but then a friend of mine make me feel uncomfortable about it. My "proof" is this: The homology ...
2
votes
1answer
34 views

Künneth Theorem For Product of Projective Spaces

I have a question about an argument using the Künneth Theorem for cohomologies: Let $X, Y$ topological spaces and $X$ has finite homology groups. Then Künneth provides for every degree $n \ge 0$ we ...
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0answers
11 views

Homogeneous space construction theorem and expected dimensions of quotient maps

I am reading the notes linked here. On page 3 we read at Theorem 2 that the left coset space $G/H$ is a topological manifold of dimension $dim(G)-dim(H)$. Here $G$ is a Lie group and $H$ a closed ...
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1answer
52 views

Homotopic maps induce the same homomorphism for reduced homology groups

This is an exercise from Hatcher's Algebraic Topology book: Exercise 2.1.13, pg. 132: Verify that $f≃g$ implies $f_∗=g_∗$ for induced homomorphisms of reduced homology groups. The formal version of ...
2
votes
0answers
24 views

Order relation between cohomology groups.

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
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votes
2answers
27 views

Betti number of the cut torus

I've read that the cut torus (the torus with a disk chopped off the end) has homotopy type of the figure eight curve. Why is this and what is the Betti number of the cut torus?
5
votes
1answer
45 views

Homology change by removing a simplex and its face

Let $K$ be an abstract simplicial complex and $\alpha,\beta$ be a $k-1$-dimensional and $k$-dimensional simplex of $K$ respectively, such that: $\alpha$ is a proper face of $\beta$ $\alpha$ is not a ...
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vote
1answer
40 views

How to calculate $H_n(\ast)$?

Let $X=\{\ast\}$ be the space with just one point. I want to calculate $H_n(\ast)$. The n-th homology. First I have to give $C_n(X)$. In my lecture it was stated, that $C_n(X)=\mathbb{Z}$. I want to ...
2
votes
2answers
50 views

Definition of graded abelian group

I am reading Homology Theory by Vick, and in this book a graded abelian group $G$ is defined to be a “collection of abelian groups {$G_i$} indexed by the integers with component-wise operation”. What ...