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

1
vote
0answers
19 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
20 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
24 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
11 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
34 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
34 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
25 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
32 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
54 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 ...
1
vote
0answers
21 views

Injectivity of a map from a homotopy set to a homology group

Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact. (A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto ...
1
vote
1answer
23 views

Mapping torus of Klein bottle, from discussion in Hatcher p. 152.

At the very bottom of page 151 to the top of 152 in Algebraic Topology by Hatcher, it says In the case of the mapping torus of a reflection $g:S^1\to S^1$, with $Z$ a Klein bottle, the exact ...
0
votes
1answer
25 views

Basic idea for finding critical point via Morse theory

Please what is the basic idea for finding critical point via Morse theory and critical groups? Thank you
3
votes
1answer
55 views

Exercise 1.1.3 in Charles Weibel’s book “An Introduction to Homological Algebra”

I am trying to teach myself some homological algebra and I got stuck right at the start with Exercise 1.1.3 from the book “An Introduction to Homological Algebra” by Charles Weibel. Exercise 1.1.3 ...
1
vote
2answers
69 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
5
votes
0answers
185 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
4
votes
1answer
29 views

Linearly Independent Curves on Genus g Surface

I have seen the following claim: Let $\gamma_1, \gamma_2, \cdots, \gamma_g$ be a collection of $g$ non-intersecting, simple closed curves on a genus $g$ surface, $\Sigma$. Then the $\gamma_i$ are ...
2
votes
1answer
52 views

Do any 2 morphisms from objects $X$ to $Y$ define a chain homotopy equivalence?

I was curious about one thing: Let $A$ and $B$ be abelian categories with enough projectives, let $X$, $Y$ be objects in $A$ and let $P_{\bullet} \rightarrow X$, $P'_{\bullet} \rightarrow Y$ be ...
1
vote
0answers
60 views

How to prove a direct sum?

$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint. $i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by ...
0
votes
1answer
68 views

Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
0
votes
1answer
34 views

cells of quotient CW complex

Let $X$ be a CW complex and $Y$ a CW subcomplex. If $X$ has no cell of dimension $n$, for some $n>0$, then $X/Y$ has no cell of dimension $n$. Is it true? Why?
0
votes
0answers
73 views

Question about Property of Homology

I have this theorem, with the proof, but i dont understand, why they prove that $i_{1_*}, i_{2_*}$ are injective, we have that $i_{j_*},j=1,2$ are induced by an inclusion it is injective, so they are ...
2
votes
3answers
59 views

homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
5
votes
1answer
83 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
1
vote
1answer
26 views

Proof of the h-cobordism theorem

I am currently learning Morse theory. Having read Milnors "Morse Theory" I am now studying his "Lectures on the h-cobordism theorem". I have also read parts of "Lectures on Morse Homology" by Banyaga ...
1
vote
1answer
33 views

Finding generators of homology groups

Take the simplicial complex with vertices {a,b,c} and edges {ab, ac, bc}. In other words, a circle. If I build a chain complex, and make the matrix of my differential, I get that the kernel of ...
6
votes
1answer
107 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
2
votes
1answer
149 views

Homotopy invariance in homology

i have this from Hatcher's book "Algebric topology" And i don't understand why we have $i-1$ in $(-1)^{i-1}$ and strict inequality in $P\partial(\sigma)$ ? Please. Thank you.
1
vote
1answer
44 views

Property of Homology: group isomorphism

I have this proposition, but I don't understand how they use the axiom 5, since in the axiom 5; $f,g: (X,A)\rightarrow (Y,B)$ and in the theorem we have $f:(X,A)\rightarrow (Y,B)$, $g:(Y,B)\rightarrow ...
0
votes
1answer
44 views

A short exact sequence

I have this proposition, and I don't understand how to do to obtain the short exact sequence: where axiom 4 is:
4
votes
2answers
124 views

Property of homology

I have this proposition, and I have two questions: 1) Why $H_k=\text{Im} i_*\oplus \ker r_*$ ? 2) Why $j_*: \ker r_*\rightarrow H_k(X,A)$ ? Edit: For the second, I try the 1th theorem of ...
1
vote
2answers
67 views

Property of excision of Homology

Please what is the difference between these two excision property: Let $X$ a topological space, $A$ a sub-space of $X$ and $U\subset A$ such that $\overline{U}\subset \stackrel{\circ}{A}$ . The ...
1
vote
0answers
19 views

Dold's proof of equivalence singular and cellular homology

I would like to ask for some help understanding a claim in Dold's proof of the equivalence of cellular and singular homology. The point is that I don't get why $\delta_n=j_*\delta_*$ where: ...
1
vote
0answers
22 views

Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
4
votes
1answer
59 views

Mayer-Vietoris where $A\cap B$ bounds $A$ and $B$

So I'm a bit confused about how the Mayer-Vietoris Sequence works. I thought that one of the times when it is useful is when we choose $A$ and $B$ such that $A\cap B$ is homotopic to the boundaries of ...
0
votes
0answers
39 views

Question about a theorem from Chang's book: Methods in Nonlinear analysis

I have this this theorem from Chang's book: Methods in Nonlinear analysis, with it's proof, but i don't understand it, for example what it means $K(f_{\sigma_i})$ ? Please help me thank you
0
votes
0answers
84 views

Question about Homology from the Chang's book: Methods in Nonlinear analysis

In the K.C.Chang's book in page $336$ of the book this corollary without prove there is a theorem before it but I don't know if it is a corollary of this theorem, how I can prove this ...
0
votes
1answer
36 views

Question about Chains complexes

I have $\mathcal{U}=\lbrace X-U, A\rbrace$ such that $\overline{U}\subset \overset{º}{A}$ and $X=\overset{º}{(X-U)}\cup \overset{º}{A}$ where $X$ is a topological space, $A$ is a subset of $X$ and ...
7
votes
1answer
147 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
1
vote
0answers
44 views

Cup products and cross products

So, I am trying to compute some products on chains and their duals, but I have difficulties in understanding some operations. The cup product of cochains is quite easy to understand, especially when ...
0
votes
0answers
37 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
1
vote
1answer
45 views

Introduction to Localization of Topological Spaces

I am trying to learn localization of topological spaces but am not sure where to start. Can anyone recommend some introductory materials? It would be great if it contains detailed motivations, ...
3
votes
0answers
36 views

Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
2
votes
1answer
45 views

An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
0
votes
1answer
31 views

Proof for Homologous cycles

Prove that two cycles that surround the same holes differ by a boundary i.e. the relation for calling two cycles homologous as mentioned here. ...
1
vote
0answers
48 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, ...
3
votes
1answer
58 views

Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
4
votes
1answer
55 views

Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
1
vote
1answer
55 views

Definition of boundary in a topological invariant way

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, ...
0
votes
1answer
72 views

Homology of 3-sphere minus an embedding of $S^1 \times \mathbb{D}^2$

I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. ...