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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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Cohomology group for trivial group

Let $G$ be a group, $A, B$ be a $G$ - modulo. We can define the n-th Cohomology group of $G$ with coefficient in $A$. $$H^n(G,A) =\text{Ext }_G^n(\mathbb{Z},A)$$ And the n-th Homology group of $G$ ...
Kongca's user avatar
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Is this group the universal central extension?

Let $G$ be a connected Lie group and $H^2 (G;\mathbb{R})\cong \mathbb{R}=\langle [C]\rangle$ where $C:G\times G\to \mathbb{R}$ is a non-trivial two-cocycle. We can define the group $\hat{G}_C :=G\...
Mahtab's user avatar
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-2 votes
0 answers
35 views

How to compute $H^2 (\mathbb{R})$? [closed]

The Whitehead's lemma says if $\mathfrak{g}$ is a finite-dimensional semi-simple Lie algebra, then $H^k (\mathfrak{g};\mathbb{V})=0$ for all $k>0$. My question is that is $\mathbb{R}$ (as a Lie ...
Mahtab's user avatar
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2 votes
1 answer
77 views

Derivation of cellular boundary formula

I was working through Hatcher's derivation of the cellular boundary formula and was trying to fill in all the nitty gritty details. One I thing I could not understand is how to choose the appropriate ...
the_dude's user avatar
  • 596
3 votes
1 answer
49 views

Can two different spin structures on a manifold induce the same spin$^c$ structure?

Let $(M,g)$ be an oriented Riemannian $n$-manifold with transition functions $g_{\alpha\beta}:U_{\alpha \beta}\to SO(n)$. A spin structure on $M$ is a lift of $g_{\alpha\beta}$'s to functions $\tilde{...
user302934's user avatar
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3 votes
1 answer
63 views

Homeomorphism of $[0,1]^m$ with subspaces of $S^n$

I was reading the Borsuk-Ulam theorem, which states that there is no continuous map $f$ from $S^2$ to $S^1$ which satisfies $f(-x)=-f(x)$. One question came to my mind: Is there any subspace of $S^n$ ...
mahdi meisami's user avatar
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For any $\alpha \in \pi_6 {(\mathbb{C}P^2 \vee S^2)}$, $\mathbb{C}P^2 \vee S^2 \cup_{\alpha} D^7$ has the same cohomology algebra over a field.

In the book Rational Homotopy Theory by Y. Felix, S. Halperin, and J. Thomas. Exercise 4.3 asks the reader to prove that for any $\alpha \in \pi_6 {(\mathbb{C}P^2 \vee S^2)}$, the space $X_{\alpha}=\...
Jiahao Li's user avatar
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1 answer
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${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$.

For a field $k$, I am calculating ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$, where $\epsilon^2 = 0$. However, there seems no complete explanation as far as I checked. ...
Pierre MATSUMI's user avatar
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Compute Homology [closed]

I am solving the following question. I want to make sure about my answer: Given a simplicial complex with vertices $𝑣_1$,$𝑣_2$,$𝑣_3$,$𝑣_4$,$𝑣_5$,$𝑣_6$,$𝑣_7$,$𝑣_8$, , the edges and a single ...
tony's user avatar
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2 votes
1 answer
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For a compact 4-manifold, no 2-torsion in $H_1(M;\Bbb Z)$ implies no 2-torsion in $H_n(M;\Bbb Z)$ for all $n$

Let $M$ be a topological compact connected oriented 4-manifold with nonempty boundary, and suppose that each boundary component of $M$ is a rational homology 3-sphere. Is it true that if $H_1(M;\Bbb Z)...
user302934's user avatar
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Calculating homology of cobordism of 3-manifolds from Kirby diagram

I've been reading Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz, and have gotten stuck on the following exercise on p. 44. Below $Y_1, Y_2$ are closed 3-manifolds, and $Q$ ...
Hrhm's user avatar
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The definition and properties of the restriction of integral (rational) cohomology to a subspace within the ambient space.

This seems to be a fairly standard question, but I am somewhat confused when considering it using Čech cohomology theory. Let $X$ be a paracompact topological space admitting good cover and $Y$ a ...
msecauchy's user avatar
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1 answer
37 views

Cup products in the rational cohomology of products of spaces

I'm starting to learn cup products and I have been stuck at a problem. I need to find the cup product structure in the rational cohomology ring of $X = S^3 \times T^3$, where $S^3$ is the $3$-sphere ...
DavidChi's user avatar
1 vote
1 answer
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Rational homology sphere has the same cohomology ring as sphere

Let $M$ be a closed, $n$-dimensional manifold such that $H_0(M;\mathbb{Q}) = \mathbb{Q} = H_n(M;\mathbb{Q})$ and $H_i(M;\mathbb{Q}) =0$ for all $1 \le i \le n-1$. Then $M$ is a rational homology ...
ShamanR's user avatar
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9 votes
1 answer
174 views

Contradiction in Computation of Homology Groups of the Mapping Class Group of a Surface?

One of the two main results of a paper by Nathalie Wahl on homological stability of the mapping class group of a surface is the following: Theorem 1.2 The map $H_*(\delta_g) : H_*(\Gamma_{g,1};\mathbb{...
jasnee's user avatar
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1 answer
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Definition of homology group as quotient in chain complex

I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
Flynn Fehre's user avatar
1 vote
1 answer
35 views

why is $\sum_{j=1}^{m}a_j\partial(\sigma_j)=\sum_{i}(-1)^i \sigma|[v_0,...,\hat{v_i},....,v_n]$

There is some confusion regarding the boundary map According to Gereon Quick book (see page no:$44$) boundary operator is define by $$\partial_n:S_n(X)\to S_{n-1}(X)$$, $$\partial(\sum_{j=1}^m a_j\...
jasmine's user avatar
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-1 votes
0 answers
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Cohomology groups of the three sphere [duplicate]

I look for an a reference for the dimension of the de Rham cohomology groups $H^*$ of the three sphere $S^3$. I would guess that it is true that $$ dim(H^0) = 1, ~~ dim(H^1) = 0, ~~ dim(H^2) = 0, ~~ ...
Zoltan Fleishman's user avatar
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0 answers
40 views

An explicit sheaf/Cech cohomology computation

There is a sheaf cohomology computation involved in a problem I'm working on, and I'm aiming to try to solve it for the simplest non-trivial case. But I'm struggling even for that simple example. Let $...
Paul Cusson's user avatar
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1 vote
0 answers
48 views

How to get the algebraic expression of the prism operator in any dimension?

I am reading Hatcher's Algebraic Topology and am curious about how to design the algebraic expression of the prism operator even if I know the intuition is to create a cylinder such that the boundary ...
Tsao's user avatar
  • 19
2 votes
1 answer
55 views

Reference for wiki claim about degree of spheres

I have seen the following claim on the wikipedia page for 'vector field' about degree theory on vector fields. The index is not defined at any non-singular point (i.e., a point where the vector is ...
JDoe2's user avatar
  • 766
1 vote
2 answers
109 views

If $f:X \to Y$ induces isomorphism in cohomology, then it induces isomorphism in homology

Let $f:X \to Y$ and $G$ a group (I'm interested specifically in the case $G=\mathbb{Q}$) such that $f^\ast:H^q(X;G) \to H^q(Y;G)$ is an isomorphism for all $q$. It it true that $f_*:H_q(X;G) \to H_q(...
marc's user avatar
  • 641
1 vote
1 answer
89 views

Question about the proof of Serre finiteness theorem

I'm studying the Serre finiteness theorem: Theorem The homotopy group $\pi_i\left(S^n\right)$ is finite for all $i$ except for $i=n$ and if $n$ is even for $i=2 n-1$, when it is finitely generated of ...
marc's user avatar
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1 vote
0 answers
51 views

Computation of the Rings $H^*(K(\mathbb{Z}, n) ; \mathbb{Q})$

I'm studying Fomenko-Fuchs lecture 26.2 and I'm studying the following Theorem at page 371: Theorem. $$ \text{n odd} \implies H^*(K(\mathbb{Z}, n) ; \mathbb{Q})=\Lambda_{\mathbb{Q}}(x), \text{dim}x=n; ...
marc's user avatar
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1 vote
0 answers
27 views

Question about homology of k[x]

I want to compute $Tor_1^A(k,k)$, where A = k[x],k - algebraically closed field. I know how to compute Tor functor through Koszul resolution, but I want to do it strictly through tensoring bar ...
VadimStacheff's user avatar
1 vote
1 answer
49 views

Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?

'Galois cohomology of Algebraic number fields' written by K. Haberland reads the following lemma in page 66. Let $H$ be a finite group. Let $p$ be a prime number and $H_p$ be a fixed p Sylow subgroup ...
Poitou-Tate's user avatar
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1 vote
1 answer
60 views

Question about Whitehead Tower

I'm studying Miller lectures on Algebraic Topology, and I'm stuck in Theorem 68.9 ($\bmod \mathcal{C}$ Hurewicz Theorem ): Theorem 68.9 (Mod $\mathcal{C}$ Hurewicz theorem). Assume that $\mathcal{C}$ ...
marc's user avatar
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0 votes
0 answers
29 views

Understanding cellular boundary formula

In Hatcher's book, the cellular boundary formula is defined as follows: $\textbf{Cellular Boundary Formula}$:$d_n(e^{\alpha}_n) = \sum_{\beta}d_{\alpha\beta}e_{\beta}^{n-1}$ where $d_{αβ}$ is the ...
Ubik's user avatar
  • 488
0 votes
0 answers
56 views

Induced homomorphism on homology groups

Given a topological space $X$ with subspaces $A \subset B \subset X$ the identity mapping $\text{id}: X \longrightarrow X$ induces a map of pairs $\text{id}_\#: (X,A) \longrightarrow (X,B)$. What can ...
Michael Williams's user avatar
0 votes
1 answer
117 views

Where exactly are we using the fact that $f$ is an odd function in the proof of the Borsuk-Ulam Theorem? [closed]

Consider the proposition that $f:S^n \to S^n$ is an odd map implies $f$ has an odd degree, which is essentially the proposition used in Borsuk-Ulam theorem's proof as given in Hatcher. Now one point ...
Kishalay Sarkar's user avatar
3 votes
1 answer
54 views

$\bmod \mathcal{C}$ Vietoris-Begle Theorem

I'm studying Miller Lectures on Algebraic Topology, and I'm stuck in Proposition 68.7: Proposition 68.7 (Mod $\mathcal{C}$ Vietoris-Begle Theorem). Let $\pi: E \rightarrow B$ be a fibration such that ...
marc's user avatar
  • 641
2 votes
1 answer
149 views

Is every (finite) simplicial complex the nerve of some covering?

I need to prove that for every finite simplicial complex $\Delta$ exists a Hausdorff paracompact space $X$ and a good covering $\mathfrak{U}$ of $X$ such that the nerve of the complex is $\Delta$. I ...
Juan MF's user avatar
  • 113
3 votes
1 answer
35 views

Generalizing the relative homology group of the solid torus relative to the hollow torus

In Hatcher's Algebraic Topology textbook, we are given some tools to calculate relative homology groups, the prime example of which, as I have seen, are finding the relative homology group of the ...
Cordon Smith's user avatar
2 votes
1 answer
38 views

Global dimension of a ring and Ext functor

Let $R$ be a commutative ring. The global dimension of $R$ is defined as the supremum over all cohomological dimensions (or projective dimensions) of all $R$-modules. I know that the $\operatorname{...
Conjecture's user avatar
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1 vote
0 answers
40 views

Homology group of an attaching space

Consider a cylinder without bottom nor top: $S^1 \times [0,1]$ and another circle $S^1$ on the complex plain. We may define a map of $f: z \rightarrow z^2$ on $\mathbb{S}^1$, which is a map of degree ...
Ubik's user avatar
  • 488
6 votes
0 answers
122 views

Did the Descartes-Euler conjecture influence Poincaré's theory of homology?

What is the Descartes-Euler Conjecture? all simply-connected polyhedra with simply-connected faces are Eulerian $V-E+F=2$. Additional explanation: The lemma which was falsified by the ring-shaped ...
user1274233's user avatar
2 votes
0 answers
54 views

How to set up this problem geometrically? (Hatcher AT Page 131 Problem 3)

I'm attempting to go through all of Hatcher's problems on homology. I was able to do 1, 2, 4, and 5 so far, but I don't know what he's asking for geometrically in 3. I see this thread (Hatcher ...
Nate's user avatar
  • 894
2 votes
1 answer
57 views

Spectral sequence of $\text { the fibration } E X \xrightarrow{\Omega X} X$

I'm studying Fomenko-Fuchs Lecture 22.3 and i'm stuck on the proof of the following theorem at page 333: Theorem: Let X be a topological space (with a base point), and let the space $X$ be $(n-1)$-...
marc's user avatar
  • 641
2 votes
1 answer
79 views

A quick question on equivalence between cellular and singular homology

We know that for a CW complex $X$, the cellular and singular homology groups are isomorphic: $H_n (X) \cong H_n ^{CW} (X)$ for each $n \in \mathbb{N}$. My question is the following: suppose $Y$ is ...
the_dude's user avatar
  • 596
0 votes
0 answers
44 views

Relation between two theories of degree of maps

For a holomorphic map between 2 Riemann surfaces, we can define its degree to be the sum of ramification index of all preimages of a point, or simply the cardinality of the preimage of a regular(not ...
minukesis's user avatar
2 votes
0 answers
36 views

Cohomology class of automorphism group of Galois form

Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
gimothytowers's user avatar
2 votes
0 answers
49 views

Definition of $E^{\infty}_{p,q}$ in spectral sequence $\{E^r,d^r\}_{r\in\mathbb{Z}}$

Let $\{E^r,d^r\}_{r\in\mathbb{Z}}$ a spectral sequence, that is $E^r$ are differential bigraded $R$-modules $E^r=\bigoplus_{p,q} E^r_{p,q}$ with $d^r$ differential of degree $(-r,r-1)$ (that is $d^r=\...
marc's user avatar
  • 641
0 votes
0 answers
43 views

How to systematically compute the quotient of free abelian groups (for simplicial homology)

I recently read an article that states that the quotient group G/H = <a+b, c>/<a+b-c, a+b+c> is equal to the quotient group <‎c>/<2c>. From what I understand, the fact that a+b-...
Number23's user avatar
1 vote
0 answers
15 views

Relation between $L^2$-norm of harmonic forms and mass in some other (co-)homology theory

Let $(M,g)$ be a compact Riemannian manifold. Let $[\omega] \in H^k_{dR}(M)$ be a deRham cohomology class. Then $[\omega]$ contains a unique harmonic $k$-form, say $\omega \in \Omega^k(M)$. On the ...
user505117's user avatar
2 votes
1 answer
63 views

cohomological spectral sequence of a CW-complex

I'm studying Lecture 21.1 of Fomenko-Fuchs. Let $X$ a CW-complex with skeletons $\{X^{p}\}_{p\in\mathbb{N}}$. We consider $C=C^*(X ; G)=\bigoplus_r C^r(X ; G)$ with the differential $\delta: C^r(X ; G)...
marc's user avatar
  • 641
1 vote
0 answers
38 views

Quotient space of $S^1\times \mathbb{C}P^4$

I am interested in the quotient spaces of $S^1\times \mathbb{C}P^4$ by relation $(\zeta,[z_0,z_1,z_2,z_3,z_4])\sim(-\zeta,[\bar{z}_0,-i\bar{z}_1,\bar{z}_2,-i\bar{z}_3,\bar{z}_4])$ and relation $(\zeta,...
fasdgr's user avatar
  • 291
0 votes
0 answers
27 views

Nerve theorem and cycles deformation

Consider a finite set of point $P\subset[0,1]^{d}$. Let $r>0$, by nerve theorem, we know that, the Čech complex $C^{2r}(P)$ and $B_{2}(P,r)$ are homotopy equivalent. Furthermore, Proposition 3.2 of ...
BabaUtah's user avatar
2 votes
1 answer
77 views

The natural map from compact vertical cohomology to de Rham cohomology is not injective

Let $\pi:E\to M$ be an oriented vector bundle. In Bott-Tu's book Differential Forms in Algebraic Topology, the compact vertical cohomology $H^*_{cv}(E)$ is defined by using differential forms $\omega$ ...
user302934's user avatar
  • 1,618
2 votes
1 answer
113 views

Adem relations of Steenrod algebra

Let $\mathrm{Sq}^i$ the Steenrod operations. These are the Adem relations: when $ i<2j$ $$ \mathrm{Sq}^i \mathrm{Sq}^j=\sum_{k \in \mathbb{Z}}\binom{j-k-1}{i-2 k} \mathrm{Sq}^{i+j-k} \mathrm{Sq}^k,$...
marc's user avatar
  • 641
0 votes
0 answers
28 views

the symbol defining a cocycle ? Or rather its square?

Let S be a Riemann surface. A quasi-meromorphic function on S is a function holomorphic everywhere except finitely many points and such that locally it can be written as $re^{\phi}$ where both r and φ ...
MOHAMED BENSAID's user avatar

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