Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
4
votes
1 answer

problem 14 of section 1.2 from Hatcher

Consider the quotient of a cube $I^3$ obtained by identifying each square face with the opposite square face via the right-handed screw motion consisting of a translation by one unit in the direction perpendicular to the face combined with a…
Simon Zhu
  • 125
4
votes
2 answers

If two space X and Y have isomorphic fundamental group then when can we say that this isomorphism is induced by some map f from X to Y?

If two space X and Y have isomorphic fundamental group then when can we say that this isomorphism is induced by some map f from X to Y ? and also what can we say about this question when we take higher homotopy group in place of fundamental group ?
Shivani Sengupta
  • 611
4
votes
1 answer

Mathematics involved in Captcha Solving

I was wondering if anyone knew about Mathematics involved in Captcha Solvers. I'm not interested in spamming, I just it would be interesting to see how to classify letters. I'm certain homology is used to determine the holes of the letter. I also…
six
  • 253
4
votes
1 answer

how to represent a relative cohomology class

Let $X$ be a topological space, and $A \subseteq X$ a subspace. How to think about an element $u \in H^n(X, A)$? Is the following correct? $u$ can be represented by a function $U$ taking an $n$-cell $\sigma$ in $X$ as input and assigning an integer…
usr0192
  • 3,157
4
votes
1 answer

Application of Fixed Point Theorem

Can we prove, if $f:\mathbb{D}^2\rightarrow\mathbb{D}^2$ is a homeomorphism then $f(S^1)=S^1$ and $f(\textrm{int}(\mathbb{D}^2))=\textrm{int}(\mathbb{D}^2)$, using fixed point theorem? I have already solved this problem using fundamental groups, but…
maths_freak_007
  • 107
4
votes
2 answers

Showing that $S^2 \times S^3$ is not homotopy equivalent to $S^2 \vee S^3 \vee S^5$

I'm trying to show that $S^2 \times S^3$ is not homotopy equivalent to $S^2 \vee S^3 \vee S^5$. My argument is that removing any point from the first space leaves a space which is connected (since it is a 5-manifold, so locally looks like…
Joe
  • 1,310
4
votes
0 answers

a topological space with finite integer first cohomology group?

One more problem preparing for a PhD exam! It states "describe a space such that $H^1(X,Z)=Z_5$." I thought this was impossible by the universal coefficient theorem since $H_0(X;Z)$ is always free, and $HOM(H_1(X,Z),Z)$ is free then $H^1(X,Z)$ is…
chachoman
  • 93
4
votes
1 answer

A question about covering space

Let $p: T \to X$ be a covering and let $f:Y\to X $ be a continuous function we define $f^*T$ as $$ f^*T=\{(y,\tilde{x})\in Y\times T|f(y)=p(\tilde{x})\} $$ let $p':f^*T\to Y$ be the map given by $p'(y,\tilde{x})=y$ a) Prove that $p':f^*T\to Y$ is a…
noname1014
  • 2,513
4
votes
1 answer

A question about homology group

For a pair of spaces $X,Y$ we have $H_*(X)=H_*(Y)$. Can we necessarily find a continuous function $f$ from $X$ to $Y$ or from $Y$ to $X$, such that $f_*$ induces the isomorphism of homology group?
Summer
  • 6,893
4
votes
0 answers

Different Proof for $\mathbb{R}^m\cong \mathbb{R}^n$ if and only if $n=m$.

Using homotopy it is easy to prove that (in topology) $\mathbb{R}^n\cong \mathbb{R}^m$ if and only if $n=m$. This result seems intuitively true, but, as realized very earlier and almost everyone who tries to prove it, that the proof is not so easy.…
p Groups
  • 10,228
4
votes
1 answer

Remark in Hatcher's Algebraic Topology on Mapping Cylinders

On p2 of Algebraic Topology, Hatcher defines mapping cylinders as follows: For a map $f: X \to\ Y$, the mapping cylinder $M_f$ is the quotient space of the disjoint union $(X \times I) \cup Y$ obtained by identifying each $(x,1) \in X \times I$…
Cowabunga
  • 41
4
votes
1 answer

Compute the fundamental group

Let X be the quotient of $S^1 \times [0,1]$ by the identification $(x,y) \sim (e^{2\pi i/3}x,y), y \in \{0,1\}$. Isn't it just a rotation of the cylinder's top and bottom by the same degree? Shouldn't the fundamental group just be the fundamental…
Keith
  • 1,383
4
votes
1 answer

Odd dimensional manifold has 0 euler characteristic

I am reading Cor 3.37 of hatcher's book. This first proves that for orientable odd dimensional manifold the euler characteristic is 0, which is easy. Then for non-orientable manifold, to apply poincare duality again, he choose the coefficient to be…
user198206
4
votes
1 answer

Misunderstanding connected with the Mayer-Vietoris sequence

I tried to compute the homology groups of 2 disjoint segments $ H_k(I_1 \cup I_2; G) $ and cylinder $ H_k(C, G) $ using the Mayer-Vietoris sequence. It looks like $$ 0 \to H_1(I_1 \cap I_2; G) \to H_1(I_1; G) \oplus H_1(I_2; G) \to H_1(I_1 \cup I_2;…
Stanislav Morozov
  • 1,014
4
votes
1 answer

problem regarding fundamental and homology groups

i have completed a basic course in algebraic topology and am currently pursuing a course in homology theory.Since the day i have started a course in homology theory and got involved in finding homology groups etc,i have always faced two…
Abhishek Shrivastava
  • 854
1 2 3
…
99 100