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
0
votes
0answers
20 views

$h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$

Let $X$ and $Y$ be topological spaces and $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$. To solve this problem I ...
1
vote
1answer
28 views

A compact convex set $A$ is contained in $\mathbb{R}^n$ then for any section $x \rightarrow a_x$ there is a unique $a_A \rightarrow a_x$

A compact convex set $A$ is contained in $\mathbb{R}^n$ and $R$ is a ring with an identity. I need to show that for any section $$x \rightarrow a_x \in H_n(\mathbb{R}^n|x ; R)$$ there is a unique $a_A ...
1
vote
1answer
35 views

How do you do simplicial homology with $0, 1$ coefficients?

I want sums of integers $\sum_{i=1}^n c_i z_i, z_i \in \Bbb{Z}$ with coefficients in $\{0, 1\}$. Do I just state that all coefficients are always $0$ or $1$, or do I work with $\Bbb{Z}[\Bbb{Z}/2\Bbb{...
1
vote
0answers
22 views

Need a description of the map $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ on $2$nd Cohomology level

Let $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ be the inclusion map. I want to know what is $i^*$ on $2$nd Cohomology level. Seems it should be an isomorphism ...
0
votes
2answers
69 views

free group acting on the 3D sphere

Let's say I have a 3 dimensional sphere $S^2$ and I define the following transformation $R_1$ rotate the sphere around the north-south axis of $\frac{\pi}{2}$ (north and south are fixed points till ...
1
vote
1answer
33 views

Definition of equivariant chain complex.

What do we mean by equivariant chain complex? Is it a chain complex with some property ? I looked in many references and i did not find a definition of the expression "equivariant chain complex", I ...
3
votes
1answer
31 views

A function $f:X \to Y$ which induces isomorphisms on all homology groups but $X$ and $Y$ have non-isomorphic cohomology rings.

I am trying to find a function $f:X \to Y$, where $X$ and $Y$ are path connected, such that $f$ induces isomorphisms on all homology groups, but $X$ and $Y$ have non-isomorphic cohomology rings. It ...
4
votes
1answer
66 views

Some Homology and Cohomology Calculations for the quotient map $q:S^n\rightarrow \mathbb{R}P^n$

I am trying to calculate maps induced by the quotient map $q:S^n\rightarrow \mathbb{R}P^n$, i.e. the descriptions of the maps: $(1)q_*:H_n(S^n)\rightarrow H_n(\mathbb{R}P^n)$ $(2)q^*:H^n(\mathbb{R}P^...
0
votes
0answers
14 views

Checking for homologically non trivial cycles in the 1st homology

enter image description here The image shows the 4 equivalence classes of the 1st homology on a torus and I was wondering if there was an easy way to check, given any particular homology, whether it ...
6
votes
0answers
71 views

How to compute cohomology of $C^0(S^n,X)$?

While there are many approaches to the (singular) cohomology of free loop spaces $LX=C^0(S^1,X)$ in the literature, I can't find many results about the cohomology of the mapping spaces $C^0(S^n,X)$ ...
1
vote
0answers
32 views

GAP – compute kernel of a matrix with coefficients in a finite ring

I need to compute first cohomology group with coefficients in $(\mathbb{Z}/n\mathbb{Z})^m$ of specific finite groups. I reduced the computation of cocycles to the following problem: compute the kernel ...
0
votes
1answer
16 views

Why does the relative homology group vanish modulo the set of proper faces

Let $K$ consist of n-simplex and it's faces, let $K_0$ be the set of proper faces of K. Then why does the group $C_p(K)/C_p(K_0)$ vanish for $p<n$? Is it because there could be some elements in $...
1
vote
1answer
40 views

Prove that $f_*: H_2(T; \mathbb{Z}/2)\to H_2(K; \mathbb{Z}/2)$ is the trivial mapping.

Let us denote by $T$ the torus and $K$ the bottle of Klein. Let $f:T\to K$ be a continuous function. Prove that $f_*: H_2(T; \mathbb{Z}/2)\to H_2(K; \mathbb{Z}/2)$ is the trivial mapping. I know that ...
0
votes
1answer
44 views

To Understand The Cohomology ring $H^*(\mathbb{R}P^n;\mathbb{Z})$

I am trying to understand the Cohomology ring structure of $\mathbb{R}P^n$ with $\mathbb{Z}$ coefficients using the Coholomogy ring of of $\mathbb{R}P^n$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients. ...
5
votes
1answer
43 views

Any embedding $\mathbb{R} P^n\hookrightarrow \mathbb{R} P^{\infty}$ induces a surjection of the singular cohomology rings.

On page 79 of "Vector Bundles and K-Theory" Hatcher makes use of the claim in the title where he considers cohomology with integer or $\mathbb{F}_2$ coefficients. Why is this claim true? I know ...
1
vote
1answer
47 views

Exercise 1.1.2, Weibel

From Weibel's An introduction to homological algebra: Exercise 1.1.2 Show that a morphism $u:C\rightarrow D$ of chain complexes sends boundaries to boundaries and cycles to cycles, hence maps $...
1
vote
0answers
34 views

What is the differential $d^0(f) : C^0(G, A) \to C^1(G, A)$ knowing the general formula?

Romyar Sharifi's Lecture Notes It's from the first differential formula appearing on page 7 and that formula is: $$ d^i : C^i(G, A) \to C^{i+1}(G, A), \\ d^i(f)(g_0, \dots, g_i) = g_0 f(g_1, \dots, ...
1
vote
2answers
43 views

Example of a non surjective and non injective induced homomorphism on $H_1$.

I'm trying to solve the following problem from a past qualifying exam on algebraic topology: Give an example of a space $X$ and a finite sheeted, connected cover $p:Y\to X$ such that the induced map $...
0
votes
2answers
40 views

[Example 3.11, Hatcher]Cellular coboundary maps with $\mathbb{Z}$ coefficients are zero then so are those with arbitrary coefficients.

In Example 3.11 of Hatcher's Algebraic Topology, it says that All cellular boundary maps for $T^n$ with $\mathbb{Z}$ coefficients must be trivial, otherwise the cohomology groups would be smaller ...
1
vote
2answers
34 views

Equivalence classes and orientation

From Elements of Algebraic Topology by Munkres, Let $\sigma$ be a simplex. Define two orderings of its vertex set to be equivalent if they differ from one another by an even permutation. If $\dim \...
0
votes
1answer
35 views

Question about calculation in Mayer-Vietoris sequence

Suppose I want to to compute the homology group of $X=S^1 \times (S^1 \vee S^1)$, which can be seen as two toruses piled together. Now, by Mayer-Vietoris sequence, I can split $X$ into two toruses $U,...
2
votes
2answers
45 views

To understand a long exact sequence related to Cohomology

I am trying to understand a long exact sequence from some notes. $I=[0,1] $ and $X$ any topological space, We look at the long exact sequence for the pair $(I\times X,\partial I\times X)$ $\dotsb\...
4
votes
0answers
90 views

Hatcher's proof of Leray–Hirsch theorem

Questions: I think the definition of $\Phi$ relies on the assumption (b), but the left and right $\Phi$ from the commutative diagram don't have such an assumption. I wonder how is the left and right ...
6
votes
1answer
76 views

Homology of dihedral groups of even degree

I'm looking for a reference or an idea about how to calculate the homology groups of dihedral groups $D_{2n}$ with integer coefficients or any abelian group when the degree $n$ is even. Here https://...
1
vote
0answers
60 views

Does a dominant morphism induce a surjection in (first) homology?

Let $X$, $Y$ be complex normal quasi-projective varieties and let $f\colon X\to Y$ be a dominant morphism. I would like to ask if the following statements are true: $f_{*}\colon H_{1}(X; \mathbb{Q})\...
3
votes
2answers
44 views

(co)homology functor and basepoint-preserving homotopy class

We can use $h^n(X)=\langle X,K(G,n)\rangle $ to define a reduced cohomology theory. I wonder if we can use the basepoint-preserving homotopy classes $\langle -,-\rangle $ to define homology? And I ...
4
votes
0answers
85 views

Two Definitions of Cohomologies

There are two definitions of cohomologies I've found so far: The first (very good to understand) definition is via coboundaries and is very often written as $$ H^p = {Z^p}/{B^p} ,$$ and the second one ...
2
votes
1answer
63 views

Confusion about the top homology group of a compact manifold.

I know that if the manifold is compact, then all of its homology groups are finitely generated. But on the other hand, we know (for example Hatcher 3.26) that if the manifold is closed and orientable, ...
2
votes
1answer
37 views

What’s the quotient space of this 2-simplex?

Let $\bigtriangleup ^2$ denotes the standard 2-simplex in $R^3$. I am wondering what is the quotient space of this simplex if we identify all three edges to a single edge? I guess it would be 2-sphere ...
1
vote
0answers
9 views

Homotopy operator commutes with sections

Exercise 11.10 on page 122 of Bott and Tu asks us to show that, if $s:M\to E$ is a section (of a sphere bundle, in context), then $s^* K = K s^*$, where $K$ is the homotopy operator for the Cech ...
1
vote
0answers
28 views

Associator of a 2-cocycle is a 3-cocycle

Let $A$ be a $k$-algebra with underlying vector space $V$ and let $F_1:V\times V\to V$ be a bilinear map. Let $A:V\times V\times V\to V$ be the associator of $F_1$, i.e. $A(a,b,c)=F_1(F_1(a,b),c)-...
-2
votes
1answer
61 views

find a cochain map $ψ: A• → B•$ with a condition that $ψ^i:A^i→B^i$ is an inijective [duplicate]

I try to find an example for a cochain map $ψ: A• → B•$ that exists : $ψ^i:A^i→B^i$ is an injective map (for i>=0) but $ψ∗ :H^k(A∙)→H^k(B∙)$ is not a injective map (for k>=0)? I use the fact that a ...
2
votes
1answer
39 views

Homological groups of the $T^2 = S^1 \times S^1$ and quotient out the circle $S^1 \times \lbrace x \rbrace$ for some point $x \in S^1$

Homological groups of the $T^2 = S^1 \times S^1$ and quotient out the circle $S^1 \times \lbrace x \rbrace$ for some point $x \in S^1$ I am trying to calculate the homology groups of this space, but ...
1
vote
2answers
50 views

Homology groups of the space obtained from $\mathbb{D}^2$ by first deleting the interiors of two disjoint subdisks in the interior of $\mathbb{D}^2$

Homology groups of the space obtained from $\mathbb{D}^2$ by first deleting the interiors of two disjoint subdisks in the interior of $\mathbb{D}^2$ and then identifying all three resulting boundary ...
0
votes
0answers
16 views

Relative Maslov Index as a Homomorphism on Relative Homology

Let $X,w$ be a symplectic manifold and $L\subset X$ a Lagrangian submanifold. Let $J$ be an $w$-tame almost complex structure. I need to understand how to think of the Maslov index as a function $\mu:...
2
votes
2answers
74 views

Calculating homology objects in this chain complex

Problem: Let $R = k[x]/(x^2) $ where $k$ is a field and consider the chain complex $$C : \qquad 0 \xrightarrow{d_2} R \xrightarrow{d_1} R \xrightarrow{d_0} 0 $$ where $d_1 : R \rightarrow R: f \mapsto ...
0
votes
1answer
31 views

Twisted Cech Cohomology on the Circle

Exercise 10.7 in Bott and Tu involves calculating the Cech cohomology, valued in $\mathbb{Z}$, for a particular open cover of the circle. In particular, we have a good cover consisting of $U_0, U_1$, ...
2
votes
2answers
63 views

Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are homotopy equivalences.

Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are homotopy equivalences. Show that $f_∗ : H_n(X, A) → H_n(Y, B)$ is an isomorphism for all $n$. I know that there is a ...
4
votes
2answers
47 views

homology groups of the quotient of $\mathbb{S}^2$ obtained by identifying north and south poles to a point

Compute the homology groups of the quotient of $\mathbb{S}^2$ obtained by identifying north and south poles to a point. I have already calculated the homology groups in one way, but I would like to ...
2
votes
1answer
47 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{...
0
votes
1answer
31 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 ...
1
vote
1answer
29 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
votes
1answer
61 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
votes
1answer
68 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 ...
0
votes
1answer
44 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: $[\gamma].\tilde x=\tilde \gamma (1)$ ...
1
vote
3answers
48 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 $\neq \emptyset$ }$ ?
0
votes
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-...
0
votes
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
vote
0answers
23 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
vote
1answer
30 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 ...