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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2
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1answer
36 views

Quotient space of $\mathbb{S}^1\times [0,1]$obtained by identifying points in the circle $\mathbb{S}^1\times \{0\}$ that differ by $2π /3$ rotation

The quotient space of $\mathbb{S}^1\times [0,1]$ obtained by identifying points in the circle $\mathbb{S}^1\times \{0\}$ that differ by $2π /3$ rotation and identifying points in the circle $\mathbb{...
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1answer
27 views

Why $ker(i_*:H_0(\mathbb{S}^0)\to H_0(\mathbb{S}^2))$ is isomorphic to $\mathbb{Z}$?

Why $ker(i_*:H_0(\mathbb{S}^0)\to H_0(\mathbb{S}^2))$ is isomorphic to $\mathbb{Z}$? I know that $H_0(\mathbb{S}^0)\cong \mathbb{Z}^2$ and $H_0(\mathbb{S}^2)\cong \mathbb{Z}$, so the problem is ...
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1answer
26 views

Every homomorphism on cohomology groups is induced by some map on spaces

Suppose we have two topological spaces $X$ and $Y$. Let $\alpha ^* :H^n(X,\mathbb{Z})\to H^n(Y,\mathbb{Z})$ be a homomorphism. 1) Is there a continuous map $\alpha : Y\to X$ such that the induced ...
2
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1answer
57 views

Determine all abelian $G$ such that $0 \to \mathbb{Z} \to G \to \mathbb{Z_n} \to 0$ is exact - from Massey's A basic course in algebraic topology

I am self studying from Massey's A basic course in algebraic topology and I am trying do this exercise Let $A$ be an infinite cyclic group and let $B$ be a cyclic group of order $n$ with $n > 1$....
2
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1answer
58 views

Compute the homology groups $H_n(X, A)$ when $X$ is $\mathbb{S}^2$ or $\mathbb{S}^1\times \mathbb{S}^1$ and $A$ is a finite set of points in $X$.

Compute the homology groups $H_n(X, A)$ when $X$ is $\mathbb{S}^2$ or $\mathbb{S}^1\times \mathbb{S}^1$ and $A$ is a finite set of points in $X$. I am trying to understand the following solution ...
-1
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0answers
23 views

Look for a cochain map $ \psi : C_* \to D_* $ [duplicate]

I am looking for a injective cochain map $ \psi : C_* \to D_* $ such as the map {$\psi_i: C_i \to D_i$} is an injective but the map {$\psi_*: H_I(C_*) \to H_i(D_*)$} is not an injective for any $i\geq ...
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1answer
35 views

Deck or Monodromy actions to define homology with local coefficients

when we have a covering $p:Y\to X$ choose $x\in X$ and we have two actions on the fiber $p^{-1}(x)$: 1) The monodromy action of $\pi_1(X,x)$ defined as follows: $[\gamma].\tilde x=\tilde \gamma (1)$ ...
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3answers
46 views

Relative singular homology group of order zero

If $X$ is path-connected and $A \subseteq X$ then it is true that $H_0(X, A) \cong \cases {0 &if X\A = $\emptyset$ \\ \mathbb{Z} &if X\A $\neq \emptyset$ }$ ?
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1answer
19 views

Is the boundary map generated by zig-zag lemma a homomorphism between modules?

I have defined the boundary map $\partial$ successfully, but I am having trouble checking that it is a module homomorphism. In the argument, I used the surjectivity of a map to obtain a non-empty pre-...
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0answers
18 views

Group homology and finite dimensionality

When working with group homology over $\mathbb{Q}$, if a group has finite dimensional group homology, it does not imply that a subgroup has also finite dimensional group homology. Nevertheless, it is ...
1
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0answers
19 views

Why consider this homology relation on vector fields?

I am studying Turaev's work on torsions of manifolds, specifically the paper Euler Structures, Nonsingular Vector Fields, and Torsions of Reidemeister Type. (I cannot find an open-access version of ...
1
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1answer
27 views

Induced map between two product manifolds

Let $f: S^2 \times S^5 \longrightarrow S^3 \times S^4$ be a smooth map. Show that the induced map $f^*: H^7_{dR}(S^3 \times S^4) \longrightarrow H^7_{dR}(S^2 \times S^5)$ is not surjective. I know ...
4
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0answers
37 views

Isomorphism between Dolbeault Cohomology groups

Let $M^n$ be a compact Kahler manifold and consider the product complex manifold $M \times \mathbb{C}^m$. By the Leray spectral sequence associated to the projection $\pi_{\mathbb{C}^m} : \mathbb{C}^m ...
3
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0answers
68 views

Use a Mayer-Vietoris sequence to calculate $S^3 - K$ where $K$ is an embedding of $S^1$.

Use a Mayer-Vietoris sequence to calculate $H_i(S^3 - K)$ where $K$ is an embedding of $S^1$. (hint: Let $S^3 = (S^3 - K) \cup O$ where $O$ is an open tubular neighborhood around $K$ that is ...
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0answers
46 views

Does the cup product commute with Mayer-Vietoris connecting homomorphism?

I am trying to prove the following for singular cohomology with coefficients in a commutative ring $R$: Let there be excisive triad $(U\cup V;U,V)$, a subspace $W\subseteq U\cup V$ and $i:(U\cap V, ...
1
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1answer
66 views

Why is $\mathbb Z$ a $\mathbb Z[\pi]-$module?

Let $X$ be a topological space with fundamental group $\pi_1X:=\pi$. I read that to recover the cellular homology of $X$ with coefficients in $\mathbb Z$ from the homology of $X$ with local ...
3
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0answers
39 views

fundamental group of the union of two cylinders

Let $X:=\{(x,y,z)\in \mathbb{R}^3: (x^2+y^2-1)(x^2+z^2-9)=0\}$. I would like to compute the fundamental group of $X$. The plot of $X$ is this. My attempt: The space $X$ is homotopy equivalent to ...
3
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1answer
52 views

“Commutativity” of Tor functor

In the book I'm reading now the author says that it's easy to prove the equality $\operatorname{Tor}_i(A, B) = \operatorname{Tor}_i(B, A)$ by doing these manipulations: (i) Let's consider projective ...
1
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1answer
36 views

Boundary operator: what does it mean?

I'm getting some confusion in simplicial homology...Take a very simple example, a (solid) tetrahedron: Following the well known property that "the bounday of a boundary is zero", we would end up with ...
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0answers
14 views

Homomorphism between graded algebras

I have a commutative cochain algebra $(A,d)$ with $H^0(A)=k$ and $H^1(A) =0.$ Choose $m_2:(\bigwedge V^2,0)->(A,d)$ so that *$H^2(m_2):V^2->H^2(A)$ is an isomorphism. Note that $H^1(m_2)$ is ...
3
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1answer
51 views

Homology and cohomology of manifolds

This is a question from Massey's Singular Homology Theory. It reads: Let $M_1$ and $M_2$ be closed orientable $n$-manifolds, and let $f : M_1 \to M_2$ be a continuous map such that $f_* : H_n(M_1) \...
0
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1answer
38 views

Is the following relation correct?

Using Kunneth theorem I would like to compute the following 4th homology: $ H_4(S^2\times S^3/\mathbb{Z}_k,\mathbb{Z})\simeq H_4(S^2,\mathbb{Z})\otimes H_0(S^3/\mathbb{Z}_k,\mathbb{Z})\oplus H_3(S^2,\...
1
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1answer
44 views

Is $H_n(X,A)$ naturally isomorphic to $\tilde{H}_n(X/A)$?

It is well known that under some assumptions on a pair $(X,A)$ of a topological space and a subspace we have $H_n(X,A)\simeq \tilde{H}_n(X/A)$. Such an assumption can be for example that $A$ is closed ...
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0answers
13 views

Construction of minimal Sullivan model

I have read a lot of papers about constructing a minimal model for a dga $(A,d)$, but still struggle to summarise the minimal data I need to construct the minimal model. Can someone help! Also, what ...
2
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1answer
43 views

What happens if we remove the point of origin from the moebius band?

What happens if we remove the point of origin from the moebius band ( using the deficion of the moebius band in $[0,1]\times[0,1]$)? what topological space do we get? I know that if we remove the ...
2
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1answer
41 views

Why $H_1(SX)\cong \widetilde{H_0}(X)$?

Why $H_1(SX)\cong \widetilde{H_0}(X)$? Suppose that $X$ is a topological space and $SX$ is the suspension of this space, so if we take as open $U=SX-\{\overline{(x,1)}\}$ and $V=SX-\{\overline{(x,0)}\...
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0answers
13 views

E3 Algebras when involving the complex numbers

The definition of $E_3$-algebra that I am familiar with involves operations parametrized by certain sets of embeddings of disjoint unions of $\mathbb{R}^3$ into itself. In this paper http://people....
2
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1answer
67 views

If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ then $f_*=g_*:H_n(X,A)\to H_n(Y,B)$. [duplicate]

If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ such that $H(a,t)\in B$ for all $a\in A$ and $t\in [0,1]$, then $f_*=g_*:H_n(X,A)\to H_n(Y,B)$. This question ...
1
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1answer
59 views

Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$.

Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$. We have a short exact succession $$0\to S_*(x_0)\to S_*(X)\to S_*(X,x_0)\to 0$$ ...
2
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1answer
30 views

$\Theta$ is induced by the inclusion map $Z_p\otimes Z_q' \to (C\otimes C')_{p+q}$?

Let $C$ and $C'$ be two chain complexes and $C\otimes C'$ their tensor product. Then define the map \begin{align} \Theta : H_p(C)\otimes H_q(C')&\to H_{p+q}(C\otimes C')\\ {[z_p]}\otimes {[z_q']}&...
6
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1answer
63 views

Interesting topological spaces to calculate the homology groups.

Interesting topological spaces to calculate the homology groups. I am calculating homology groups of several topological spaces to learn and I have already calculated the homology groups of $\mathbb{...
0
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2answers
47 views

If $\mathbb{Z}≅\widetilde{H_0}(\mathbb{R}P^2)⊕\mathbb{Z}$ then $\widetilde{H_0}(\mathbb{R}P^2)=0$?

If I were to calculate $H_3(S^3\mathbb{R}P^2)$ where $SX$ is the suspension of the topological space $X$, then I get to $H_3(S^3\mathbb{R}P^2)=H_2(S^2\mathbb{R}P^2)=H_1(S\mathbb{R}P^2)=\widetilde{H_0}...
1
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1answer
31 views

Why is $H_i(E_+^n,S^{n-1})\cong H_{i-1}(S^{n-1})?$

Let $E_n^+\subset S^n$ be the upper hemisphere of $S^n$. Why does the fact that $E_+^n$ is contractible imply that in homology $H_i(E_+^n,S^{n-1})\cong H_{i-1}(S^{n-1})?$ I know that if a space is ...
0
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0answers
26 views

Barycentric subdivision and chain maps

In Rotman's An Introduction to Algebraic Topology, 4th printing, pp. 113, he provided a definition of barycentric subdivision in a convex set $E$: Let $E$ be a convex set. Then barycentric ...
1
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1answer
49 views

Why is $H_1(S^0)=0$?

Why is $H_1(S^0)=0$? We have that $S^0=\{a,b\}$ two points right? So we have singular simplices of the form $\Delta_1 \mapsto$ a and $\Delta_1 \mapsto b$ which are non-zero no? So why is the homology ...
0
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0answers
15 views

Induced map $f_*:H_p(A,B)\to H_p(C,D)$ between relative homologies?

I understand that a map between simplicial complexes induces a map in homology. But why and how does a map $f : (A,B) \to (C,D)$ induces a map $f_*:H_p(A,B)\to H_p(C,D)$ between relative homologies?
1
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1answer
31 views

$H_p(K)=H_p(K^{(p+1)},K^{(p-2)})$?

If I have a simplicial complex $K$ and I denote the $p$-skeleton by $K^{(p)}$, then how can I see that $H_{p}(K)=H_{p}(K^{(p+1)},K^{(p-2)})$?
0
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0answers
21 views

Confusion with differential graded algebra

I have a graded vector space $V$ and exterior algebra $\bigwedge V$. Now I have defined a differential graded algebra defined with underlying algebra $\bigwedge V$. Can someone help me to understand ...
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0answers
13 views

Boundary operator preserved under isomorphism so relative homology isomorphic.

I was trying to understand simplicial excision which sais that if L is the complex we get from excising $U$ from $K$ and $L_0$ is the complex obtained by excising $U$ from the subcomplex $K_0$ then $...
1
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1answer
29 views

Splitting induces isomorphisms

Let $M^n$ be a closed topological manifold. For $1 \leq q \leq n$, we have the following short exact sequence: $$ 0 \longrightarrow \operatorname{Ext}(H_{q-1}(M),\mathbb{Z}) \stackrel{\beta}{\...
1
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1answer
31 views

homology of $RP^\infty$ is a coalgebra.

Consider (co)homology with coefficients in a field $k$ of characteristic two. It is known that the cohomology ring $H^*(RP^\infty)$ is a polynomial algebra in one variable with coefficients in $k$. ...
1
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2answers
84 views

Build an example of a topological space $A$ such that $H_4(A;\mathbb{Z})=\mathbb{Z}/2$

Build an example of a topological space $A$ such that $$H_n(A;\mathbb{Z})= \begin{cases} \mathbb{Z} & \text{ if } n=0 \\ \mathbb{Z}/2 & \text{ if } n=4 \\ ...
1
vote
1answer
20 views

Confusion with a step in the construction of chain homotopy from a homotopy

The following content is from the note from MIT. Let $X,Y$ be topological space. Let $h:f_0 \simeq f_1: X \to Y$ be a homotopy between two continuous maps. The claim we want to prove is that it ...
0
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0answers
22 views

Confusion with cohomology group

I have a differential graded algebra $(A,d)$ and image $f:H^2\otimes\bigwedge^2 E \to A^8$. (the domain is not so important) Let $R\subset H^2\otimes\bigwedge^2 E$ such that $f\restriction {R}$ ...
4
votes
1answer
70 views

Isomorphism in all homology and cohomology groups

Let $M^n$ be a closed, connected and orientable topological manifold of dimension $n \geq 2$ and let $f : M \to M$ be a continuous map. Assume that $f_* : H_n(M) \to H_n(M)$ is an isomorphism. The ...
3
votes
0answers
38 views

Homological algebra without homological grading

Suppose we have a complex without a homological grading (i.e. an abelian group). We can speak meaningfully about there being a differential $d$ on this group such that $d^2=0$. Is this creature then ...
0
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0answers
31 views

Is this a finite extension?

Let $G$ be a group, $H$ a $2$-index subgroup of $G$ and $N$ a normal subgroup of $G$. I want to understand why if $H_{1}(N; \mathbb{Q})$ is infinite dimensional, so is $H_{1}(N\cap H; \mathbb{Q})$. ...
1
vote
1answer
22 views

Non-degeneracy of the cup product in the middle degree

Let $M$ be an $R$-orientable, closed manifold of dimension $2n$ for some $n$. I read in a few texts that the cup product $$H^n(M;R)\times H^n(M;R)\to H^{2n}(M;R)$$ is non degenerate, as a result of ...
0
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0answers
43 views

Question regarding pushforward of homology under projection

Let $X$ and $Y$ be two compact connected oriented manifolds of real dimension $m$ and $n$, respectively. Let us assume that these are also smooth projective complex varieties (the reason to be clear ...
0
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0answers
30 views

Confusion with example from a paper

Let $X$ be connected 8-manifold, with $H^3=0$ (3rd cohomology group), and $a,x_1,x_2,x_3\in H^2(X)$ such that $ax_i=0$. If $\alpha,\beta_i\in \Omega^2(X)$ represent $a$ and $x_i$ respectively and $\...