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 [tag:homological-algebra] for more abstract aspects of (co)homology theory.

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6
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1answer
53 views

$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$

For Abelian groups $\pi$ and $\rho$, what is the easiest way to see that $$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)?$$My idea is the use the natural isomorphism$$[X, K(\rho, n)] \cong ...
2
votes
1answer
24 views

Coproduct of the homology coalgebra of the sphere

Let $S^m$ be the $m$-sphere and $H_*(S^m)$ be the homology coalgebra with field coefficient. Then what is the coproduct of $ H_*(S^m) $? For $x$ the generator of $H_*(S^m)$, does $$ \Delta_*x=0? $$ ...
3
votes
1answer
50 views

Geometric Interpretation of Chain Homotpy

Let $X$ and $Y$ be topological spaces. Two maps $f,g:X\to Y$ are said to be chain homotopic if for each $n$ we have a map $T_n:C_n(X)\to C_{n+1}(Y)$ such that ...
2
votes
0answers
35 views

For any Subspace $A$ of a Path-Connected Space $X$, we have $H_0(X, A)=0$.

I recently learnt about relative homologies and am wondering if the following is true: Statement: Let $X$ be a path-connected topological space and $A$ be a non-empty subspace of $X$. Then $H_0(X, ...
2
votes
0answers
24 views

What are explicit maps in the following exact sequence?

Let $G$ be a group and $M,N$ be normal subgroups of $G$ such that $G=MN$. Then there is a natural exact homology sequence $Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} H_2(G) ...
0
votes
1answer
30 views

Cohomology space

Let $M$ be a compact Riemannian manifold without boundary. a) If $M$ is a sphere, prove that the cohomology space of order $1$ is trivial: $H ^1 (M, \Bbb R) = 0$. b) If $\omega = \delta\theta$ is ...
1
vote
0answers
15 views

Restriction-Co Restriction Homomorphism

Let $G$ be a finite group and let $A$ be any $G$ module. Then it is well known that $H^n(G,A)$ is a subgroup of $\oplus_p H^n(G_p, A)$, where $G_p$ denotes a sylow $p$ subgroup of $G$. This is ...
3
votes
2answers
67 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
2
votes
1answer
46 views

Cohomology of sheaves of abelian groups on affine space

Is it true that $$\mathrm{H}^p_\mathrm{Zar}(\mathbb{A}^n_\mathbb{K}, \mathcal{F})=0$$ for any $p>1$ and $\mathcal{F}$ (non constant) sheaf of abelian groups? If not, is it true for some ...
2
votes
1answer
50 views

Question about maps to $K(G,1)$

I have an unfortunately basic confusion about two results in Hatcher's Algebraic Topology. Let $G$ be an abelian group. Theorem 4.57 specialized to $n=1$ says that there is a bijection $$\langle X, ...
-1
votes
0answers
33 views

A Question On the Herbrand Index

In the book Cohomology of Number Fields by Neukirch, Schmidt and Wingberg, I find a theorem on the computation of Herbrand index, i.e Proposition 6.1.11, which I state as follows: Let $A$ be a ...
52
votes
3answers
930 views

Topological spaces admitting an averaging function

Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$. These ...
5
votes
1answer
64 views

Dimension of $\mathbb{Q}$-vector spaces $H^m(X, \mathbb{Q})$.

Assume that you can't compute the cohomology group $H^m(X, \mathbb{Q})$ for$$X = \{(x : y : z : w) \in P^3(\mathbb{C}): xy = zw\}$$but you know Weil conjecture. By using Weil conjecture, give the ...
1
vote
2answers
77 views

Attaching maps for the CW-decomposition of the 3-torus

I want to calculate the homology of the $3$-torus via cellular homology. I figured out a CW-decomposition of the $3$-torus: $1$ $0$-cell, $3$ $1$-cell, $3$ $2$-cell, $1$ $3$-cell. So the chain complex ...
1
vote
1answer
49 views

If $r:X\to A$ is a Retraction, Then $H_n(X)\cong H_n(A)\oplus H_n(X,A)$

$\DeclareMathOperator{\im}{Im}$ Let $A$ be a subspace of a topological space $X$ such that there is a retraction $r:X\to A$ of $X$ onto $A$. Then $H_n(X)=H_n(A)\oplus H_n(X, A)$ for all $n$. ...
0
votes
0answers
28 views

De Rham Theorem for other coefficients

Are there versions of De Rham's theorem for which the coefficients are not just $\mathbb{R}$? In particular, is there a vector-valued version of De Rham's theorem?
1
vote
1answer
32 views

Volume as coboundary

For simplicity, consider simplicial homology in $\mathbb{R}^2$. It seems to me that oriented area is a cocycle, since it vanishes on simplicial cycles. The situation being Euclidean, it must ...
1
vote
1answer
42 views

Find homology $S^n-f(X)$ where f is injective

Let $f\colon X\to S^n$ be an injective function. Find the homology groups of $S^n-f(X)$ where: a. $X=S^k\sqcup S^r$ b. $X=S^k\vee S^r$ The question above gives hint to look in both ...
4
votes
0answers
36 views

Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space ...
3
votes
2answers
48 views

CW-decomposition of quotient space

Let $X$ be the space that results form $D^3$ by identifying points on the boundary $S^2$ that are mapped to one another by a $180°$-rotation about some fixed axis. I want to calculate the cellular ...
1
vote
2answers
33 views

Isomorphism of chain complexes

In my notes it says $C^{sing}_n(\sqcup_{i\in I} X_i;R) \cong {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$, where $C^{sing}_n$ denotes the n-th singular chain complex and $R$ is a ring, $S_n(X)$ is the set ...
4
votes
1answer
42 views

Relative Homology is not trivial

Let $(H_*, \partial_*)$ be a homology theory satisfying the dimension axiom. Let $A \subset S^n$ be a proper subset. Show that $H_n(S^n, A)$ is not trivial. I tried applying the long exact sequence ...
2
votes
0answers
39 views

Topological Boundary Map

In May, Concise Algebraic Topology, p. 108-109, for a cofibration $A \rightarrow X$ a "topological boundary map" is defined as the composite: $X/A \xrightarrow{\psi^{-1}} Ci \xrightarrow{\pi} \Sigma ...
3
votes
1answer
72 views

Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
2
votes
0answers
40 views

cohomology ring of cross-section space of fibre-bundles

Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be ...
7
votes
1answer
141 views

Universal coefficient theorem and multiplication on cohomology

Let $X$ be a topological space and $R$ is a commutative ring. For $H^*(X)$ we have $$0\to H^n(X,\mathbb Z)\otimes R\to H^n(X,R)\to \mathrm{Tor}(H^{n+1}(X,\mathbb Z), R)\to0.$$ Is it true that we ...
1
vote
2answers
86 views

Cohomology ring of $\mathbb RP^n$ with integral coefficient.

I know cup product structure on $H^*(\mathbb{R}P^n;\mathbb{Z}_2)= \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$. How to get $H^*(\mathbb{R}P^n;\mathbb{Z})$ from this? I have two cochain complexes for two ...
1
vote
1answer
53 views

Why does the inclusion $X\vee Y\rightarrow X\times Y$ induce an isomorphism on homology?

Let $X$ and $Y$ be nice spaces (connected, path-connected, locally contractible, etc). We do not assume they are CW complexes. There is a natural inclusion map $$X\vee Y \rightarrow X\times Y.$$ ...
2
votes
0answers
32 views

Counting the number of connected components in a complement

I have come across the following problem. Suppose you are given a closed, connected, orientable n-dimensional submanifold without boundary, in $\mathbb{R}^{n+1}$, call this submanifold $K$. Prove that ...
3
votes
1answer
50 views

cohomology ring of $S^2$ $\times$ $S^4$ and $CP^3$.

I was studying Hatcher's algebraic topology book.In page number 251,book says $S^2$$\times$ $S^4$ and $CP^3$ has same cohomology groups but they have different ring structure.I understand that they ...
3
votes
2answers
34 views

Map induced in mod $2$ cohomology of a projection $S^n \to S^n/\mathbb{Z}_2$

Consider the involution $\varphi_i \colon S^n \to S^n$ given by $(x_0, \ldots, x_n) \mapsto (x_0, \ldots, x_{i-1}, -x_i, \ldots, -x_n)$, where $0\leq i\leq n$. Let $f_i \colon S^n \to ...
2
votes
0answers
48 views

Spectral Sequence associated to a filtration abuts because we can find closed representatives

Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be ...
2
votes
0answers
26 views

What is Relator Matrix

At the third section of a paper "Computing second cohomology of finite groups with trivial coefficients" by G. Ellis et al., the authors write Suppose that $<\underline{x}\mid ...
0
votes
1answer
70 views

How can I write Klein bottle as an adjunction space?

I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I ...
0
votes
2answers
63 views

homology group of adjunction space

I start to study homology theory and i want to understand homology groups of adjunction space In this picture i can't see $V$ deformation retracts to $X$ neither intuitively nor explicitly help ...
1
vote
1answer
57 views

Induced map on the homology

Although there are good articles about this theme like induced map homology example, I would like to get a more explicit answer. I know that one way to find such a map is the following: $ f:X\to Y ...
1
vote
0answers
62 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
1
vote
1answer
67 views

Trying to make sense of this proof in Hatcher

So I'm trying to understand this proof in Hatcher's Algebraic Topology. Lemma: The composition $\Delta_n(X)\xrightarrow{\partial_n} \Delta_{n-1}(X)\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)$ is zero ...
1
vote
1answer
65 views

What is a homology of chain complex?

First of all I don't know much about homological algebra, and algebraic topology I just took a class using Kinsey book (Topology of Surfaces) which is undergraduate math class and I learned what is ...
3
votes
0answers
65 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
5
votes
2answers
66 views

How to recover the cohomology of a torus from its description of a quotient

Note: here, "cohomology" means "De Rham cohomology". I know how to compute the De Rham Cohomology of a torus $T=\left(S^1\right)^n$ using Kunneth formula. But a torus can also be obtain as a ...
2
votes
0answers
30 views

Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
4
votes
1answer
40 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
0
votes
0answers
42 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
0
votes
1answer
32 views

With field coefficients homology and cohomology coincide

With field coefficients the universal coefficients theorem takes the form: $$H^n(X;F)=Hom_{F-modules}(H_n(X;F),F)$$. Now in all computations I have seen with field coefficients we have ...
0
votes
1answer
24 views

Homology and Reduced homology coincide on non trivial pair.

In Hatcher page 118, he says that There is a completely analogous long exact sequence of reduced homology groups for a pair $(X;A)$ with $A\not = \emptyset$ ; This comes from applying ...
13
votes
0answers
113 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
2
votes
1answer
58 views

Computing Klein bottle's cohomology ring in $\mathbb{Z}$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...
8
votes
1answer
101 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
5
votes
3answers
121 views

On defining homology groups

I have been trying to understand what homology groups are "talking about," and now I am wondering if the following works as a definition of homology. But first, some illustration of what it is ...