Use this tag if your question involves some type of homology, including (but not limited to) simplicial homology, singular homology, or group homology.

learn more… | top users | synonyms

5
votes
1answer
31 views

$H_2(M)$ is free abelian for any simply connected $4$-manifold

In Naber's book "Topology, Geometry and Gauge Fields. Foundations", it is stated that for each $4$-manifold $M$ which is smooth, closed, connected and simply connected we have $H_0(M) = H_4(M)= ...
0
votes
1answer
30 views

What is $C_n(X)$?

This article says the following: Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a ...
0
votes
0answers
25 views

Applying the functor $H_*$ to the inclusion sequence $A\rightarrow B\rightarrow C$

Does applying the functor $H_*$ to the sequence of inclusions $A\rightarrow B\rightarrow C$ induce a map $\phi_3: H_*(B)\rightarrow H_*(C )$, such that if $\phi_1:H_*(A)\rightarrow H_*(B)$, and ...
1
vote
1answer
36 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
0
votes
0answers
33 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
0
votes
1answer
51 views

Homology and critical groups

I have this theoreme from the paper: J. Liu, The Morse index of a saddle point, 1989 My first question is what is $\tau$ is $\tau$ a chain ? so $I_m$ is the standard simplex ? if it is this why ...
1
vote
0answers
22 views

Barratt Whitehead Lemma, Proving exactness at the 'direct sum' module

I have been working out a proof of the Barratt Whitehead Lemma. Here it is. I had it all finished, but then I realised that I had assumed all the 'vertical' homomorphisms connecting the upper and ...
3
votes
1answer
23 views

How to show that homotopy of chain maps respects composition?

Given the homotopic pairs of chain maps $f_1 \simeq f_2 : A_* \to B_*$ and $g_1 \simeq g_2 : B_* \to C_*$, show that $g_1 \circ f_1 \simeq g_2 \circ f_2: A_* \to C_*$. $f_1 \simeq f_2$ means that ...
1
vote
1answer
60 views

On chain homotopy equivalence

I just learnt the notion of chain map and have the following question. Let $C=(C_n,\partial_n^C)$ and $D=(D_n,\partial_n^D)$ be chain complexes of abelian groups with boundary maps $\partial_n^C$ and ...
2
votes
0answers
24 views

What does the anitpodal map do to the homology groups of a sphere? [duplicate]

Let $A: S^n \to S^n$ denote the map sending a point on a sphere to the exact opposite point. Let $A_*: H_n(S^n) \to H_n(S^n)$ denote the action of $A$ on the homology group. Then $A_*$ is an ...
1
vote
1answer
42 views

Minimum number of vertices of a triangulation of the Mobius Strip

Mobius Strip and Homology Groups I've been reading Croom, Basic concepts of Algebraic Topology. I have a question. What is the minimum number of vertices in a triangulation of Mobius strip? Theorem ...
0
votes
0answers
12 views

antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
2
votes
1answer
132 views

Question about homotopy equivalence

I have this proof but I don't understand why $i\circ j$ induces a homotopy equivalence, and how to see $j_*$ is injective at the level of homology? $X$ is a Banach space
0
votes
0answers
20 views

Lefschetz Duality for Simplicial Homology

What are the duality theorems for compact, orientable simplicial complexes (possibly the triangulation of a manifold with boundary)? Is there a good way to calculate this boundary just from a list of ...
3
votes
0answers
37 views

When is $H_i(X,Y)\cong H_i(X/Y)$?

For orientable manifolds,for what $X$ and $Y\not=\varnothing$ does this isomorphism hold true?
2
votes
0answers
18 views

Fubini-Study form and homology class of curve

bit of a computation question here. Let $C$ be a (smooth) curve in $\mathbb{C}$P$^2$ (or more generally $\mathbb{C}$P$^N$) of degree $d$. Then the homology class $[C]$ is $d \cdot ...
0
votes
1answer
31 views

turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
0
votes
0answers
56 views

Finding 1st and 2nd homology groups using augmentation ideal

Let $G$ be a group and $I_{G} = ker(\epsilon)$ be the augmentation ideal where $\epsilon: Z[G] \rightarrow Z$ is the augmentation ring homomorphism $<g> \rightarrow 1$. Let $f: I_{G} ...
3
votes
2answers
55 views

Simply-connected $\mathbb{Z}_p$-homology spheres?

Let $X$ be a $\mathbb{Z}$-homology $n$-sphere, i.e., a closed manifold with $\mathbb{Z}$-homology groups of the standard $n$-dimensional sphere. If $X$ is simply-connected, it is not difficult to see ...
3
votes
1answer
40 views

Computing $H_i(\mathbb{RP}^n \times \mathbb{RP^m}; G)$

I'm trying to compute $H_i(\mathbb{RP}^n \times \mathbb{RP}^m; G)$ for $G = \mathbb{Z}, \mathbb{Z_2}$ respectively by using the cellular chain complexes. I'm not really sure how to get started, ...
2
votes
2answers
49 views

Confusion about cohomology and universal coefficients theorem.

I want to check that my understanding is correct about cohomology. Let $X$ be a topological space $G$ be an abelian group. The universal coefficients theorem, as stated in hatcher, says that the ...
1
vote
2answers
28 views

How do I compute the kernel of this map?

How do I compute $\ker{(\mathbb{Z} \otimes A \longrightarrow \mathbb{Q} \otimes A)}$ where this map comes from the short exact sequence: $0 \rightarrow Tor(\mathbb{Q}/\mathbb{Z}, A) \rightarrow ...
3
votes
1answer
61 views

Role of determinant of the matrix of any Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
0
votes
0answers
17 views

Composition of boundary homomorphism from Hatcher

We have $\partial_n ( \sigma) = \sum_{i} (-1)^{i} \sigma [ v_1, \dots, \overset { \wedge} v_i, \dots, v_n]$ and we want to show that the composition $$ \delta_n(X) \overset{\partial_n} \to ...
3
votes
2answers
58 views

Homology of $n$-sheeted covering space

Let $X$ be the Klein bottle, that is $X=\mathbb{R}^2/G$ with $$G=\langle a,b\mid a^{-1}b ab=1\rangle,$$ acting via $a: \mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto (x+1,y)$, $b: \mathbb{R}^2\to ...
1
vote
1answer
61 views

cohomology of suspension

Let $X$ be a topological space. Let $\Sigma$ be suspension. Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not? Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma ...
3
votes
1answer
46 views

homology of smash product of Eilenberg-Maclane spaces

Let $K_n=K(\mathbb{Z},n)$ be the Eilenberg-Maclane space. Prove: (1). $K_m\wedge K_n$ is $(m+n-1)$-connected. (2). $H_{m+n}(K_m\wedge k_n;\mathbb{Z})= H_{m+n}(K_m\times k_n;\mathbb{Z})$. How to ...
0
votes
0answers
11 views

stable splittings of projective space

On Hatcher's book Algebraic Topology, page 468 Prop. 4I.3, For prime number $p$, can we decompose $\mathbb{C}P^\infty$ in a similar way?
1
vote
1answer
43 views

An example on singular homology

I'm studying "Singular Homology Groups" in J. Lee's Introduction to Topological Manifolds on my own. When I try to compute some concrete examples, confusion arises: Let $p$ and $q$ be two paths in a ...
0
votes
0answers
18 views

Making a homotopy equivalence out of a pushout map involving $A \cup B = X$

Given the topological space $X$ with subspaces $A$, $B$ so that $A \cup B = X$ and the maps in the "square" of the following diagram ($i_1$, $f$, $g$, $h$) forming the pushout $Y$: I added rest of ...
0
votes
0answers
12 views

dual hopf algebras

Let $X$ be an H-space with product $\mu$. Let diagonal map $\Delta: x\mapsto (x,x)$. Let $F$ be a field. (1). Then by Kunneth formula, $H_*(X\times X;F)=H_*(X;F)\otimes H_*(X;F)$. (2). Hence $$ ...
0
votes
1answer
39 views

homology and cohomology with coefficients of ring and field [closed]

(1). Let $R$ be a ring. Let $X$ be a topological space. Then $H^n(X;R)$ is a module over $R$. Also $H_n(X;R)$ is a module over $R$.Is this statement correct? (2). Let $F$ be a field. Let $X$ be a ...
5
votes
2answers
133 views

Significance of homology groups of a topological space

I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology ...
0
votes
0answers
36 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
1
vote
1answer
32 views

Why *formal* linear combination?

In Lee's Introduction to topological manifolds on page 340 he writes that an element of $C_p(X)$ can be written as a formal linear combination of singular $p$-simplices. Similarly, on Wikipedia's ...
2
votes
1answer
30 views

Abuse of notation in relative homology theory

I am refreshing my understanding of homology theory (well, recreating from scratch really!) after a thirty year break and there's something that bugs me in how the texts I've seen write about relative ...
0
votes
0answers
10 views

Constructing an odd map on homologies from a double-inclusion of chain complexes

Given a chain of two inclusions of chain complexes $A_* \subset B_* \subset E_*$, all of which are 0 in negative degree with differentials of degree -1, the short exact sequence $0 \rightarrow B_* ...
1
vote
1answer
23 views

Homology groups of three faces with a point on the common edge removed

Consider this situation: There is an edge between two vertices, with three faces (maybe half-disks or half-squares, it doesn't really make a difference to topology as far as I know) going out from it, ...
2
votes
0answers
29 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
2
votes
1answer
47 views

the 0-th homology of a simplical complex

Let $K$ be an (abstract) simplicial complex. The claim is: $H_0(K;\mathbb{Z})$ is always nonzero. Is this possible to prove it without any "special techniques to computing homology-groups"? ...
1
vote
0answers
22 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
2
votes
0answers
23 views

$x_1$ and $x_2$ lie in the same path component iff $x_1 - x_2 \in im(\Theta)$

This is a question from Rotman's algebraic topology. Let X be a topological space and let $\Sigma$={ all paths in X }, and let $F(X)$ be the free abelian group on X and $F(X,\Sigma)$ the free abelian ...
2
votes
1answer
26 views

Morse's polynomial and Poincaré's polynomial equality

Suppose $M$ is a compact smooth manifold with Morse's polynomial $\mathcal{M}(t)$ and Poincaré's polynomial $\mathcal{P}(t)$ satisfying $\mathcal{M}(t)=\mathcal{P}(t)$ for any coefficient field ...
0
votes
0answers
15 views

Reference request for Homology Gysin sequence.

I am trying to study the Homology Gysin sequence (not cohomology). I am interested in finding references that either use, or explain the Homology Gysin sequence, especially if it gives descriptions ...
0
votes
0answers
43 views

Additivity for Relative Homology

If $(X_\alpha,A_\alpha)$ are disjoint topological pairs, is the following statement true? $$ \bigoplus_\alpha H_n(X_\alpha,A_\alpha)=H_n\left(\bigcup_\alpha X_\alpha,\bigcup_\alpha A_\alpha\right) $$ ...
0
votes
0answers
35 views

example of use of (co)homology

I'm learning cohomology, but there is very few examples in the book I'm reading. I read the definition of Ext and Tor but don't know how to use these. Are there some examples of proposition such that ...
1
vote
1answer
38 views

Explicit calculation of simplicial homology

Is it possible to calculate simplicial homology of $n$-dimensional simplex just by definition, without using homotopy invariance of homology(or it's equality to singular or cellular ones)? I've done ...
0
votes
1answer
29 views

Proving that some property on a chain complex of groups implies isomorphism between direct sums of these groups.

Let $C_*$ be a chain complex such that every $C_i$ is a torsion-free finitely generated abelian group, with $C_i=0$ for every $i<0$ and every $i>N$ for some sufficiently large integer $N$. If ...
1
vote
1answer
36 views

Homology of a 3-manifold with a solid torus attached

Let $M$ be a (connected) compact orientable 3-manifold whose boundary $\partial M$ is homeomorphic to $T^2$ (the torus). Now consider the solid torus $S=S^1\times D^2$ and choose a homeomorphism ...
1
vote
0answers
57 views

Classification of compact 3-delta-complexes made of a single simplex

With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see ...