Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

Cech cohomology cap product

This is a rephrasing of a recent post I made, then swiftly deleted. I am reading this paper, where on page 95 it is mentioned that if $X$ is a closed oriented manifold and $K \subseteq X$ is a closed ...
JMM's user avatar
  • 1,165
-1 votes
1 answer
61 views

The Motivation of Axioms for Homology [closed]

I am reading Algebraic Topology by Allen Hatcher and I'm curious about the motivation of axioms for singular homology. Singular homology is easy to be defined and understood,so I think singular ...
Tom of Halo City's user avatar
1 vote
0 answers
36 views

Projective bundle formula for Connective $K$-theory

I know how to prove that for every vector bundle $E \rightarrow X$ of degree $d$ we have: $$K^{*}\mathbb{P}(E) = K^{*}X[h]/(\lambda_{-1}[E](h)),$$ where $\lambda_{-1}[E](h) = \sum_{i = 0}^{d}(-1^{i})[\...
Eduardo4313's user avatar
1 vote
0 answers
53 views

Homological Algebra for Analysis

While studying differential forms, I encountered some concepts from homological algebra, such as (co)chain complexes, de Rham cohomology, pullbacks, and others. Is it reasonable to study the basics of ...
veirab's user avatar
  • 61
2 votes
1 answer
100 views

Higher homotopy/homology groups of smash product of a space

Let $X$ be a finite CW complex and let $X \wedge X$ denote its smash product. Given the higher homotopy and homology groups $\pi_i(X)$ and $H_i(X)$, is there a simple way to calculate $\pi_i(X \wedge ...
ShamanR's user avatar
  • 63
2 votes
1 answer
70 views

The First Singular Homology Group $H_1(X)$ and the Fundamental Group

THEOREM. Let $X$ be a topological space and let $x_0\in X$ a point. The map $\varphi\colon\pi_1(X,x_0)\to H_1(X)$ defined by $[\sigma]_{\simeq}\mapsto[\sigma]_{\sim}$ is well defined and it is a ...
Grace53's user avatar
  • 645
3 votes
0 answers
47 views

Properties of the de Rham complex for $\mathbb{R}^{3}$

I wasn't able to find a construction of de Rham complex for $\mathbb{R}^{3}$ using de Rham theorem. This is my attempt, in which I have some uncertainties. Consider $$ \Omega^{0}(\mathbb{R}^{3},\...
Matthew Willow's user avatar
0 votes
1 answer
43 views

Universal Covering Space of an $n$-dimensional CW Complex

Let $X$ be an $n$-dimensional CW-complex. Let $\tilde{X}$ be its universal covering space. I want to determine if $$H_i(\tilde{X},\mathbb{Z})=0,\,\,\,\,\, i\geq n+1.$$ I would like to say that $\tilde{...
Akhalbing's user avatar
3 votes
0 answers
39 views

Homology and Brouwer degree theory over: the quotient space obtained of the suspension of $S^1 \vee S^1$ identifying its endpoints.

Hi, everyone! I think my solution isn't the more economic. Actually, I'm not satisfied, and to be honest I think I miscalculated the celular homology of $$ X= \dfrac{(S^1 \vee S^1)\times[-1,1]}{(S^1\...
A-Train's user avatar
  • 31
0 votes
0 answers
22 views

How can both the Čech complex and the alpha complex have the same homotopy type as the union of balls if they are constructed differently?

I understand that, according to the nerve theorem, both Čech and alpha complexes have the same homotopy type as the union of balls. However, consider the following four points: A = $(1,0)$ B=$(-1,0)$,...
QuinnTheEskimo's user avatar
1 vote
1 answer
73 views

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
  • 94
0 votes
0 answers
19 views

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
  • 763
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
54 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
  • 1,630
3 votes
1 answer
65 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
0 votes
0 answers
71 views

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
0 votes
1 answer
39 views

${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ 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
2 votes
1 answer
100 views

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
  • 1,630
3 votes
0 answers
98 views

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
  • 3,405
0 votes
0 answers
32 views

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
0 votes
1 answer
40 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
58 views

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
  • 63
9 votes
1 answer
177 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
  • 2,551
0 votes
1 answer
44 views

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
37 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
  • 14.6k
-1 votes
0 answers
23 views

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
0 votes
0 answers
43 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
  • 2,077
1 vote
0 answers
49 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
112 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
91 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
  • 641
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
  • 641
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
50 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
  • 6,351
1 vote
1 answer
61 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
  • 641
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
59 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
118 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
56 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
150 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
36 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
39 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
  • 3,270
1 vote
0 answers
41 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
58 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
37 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

1
2 3 4 5
112