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
3
votes
1 answer

Question 13 in section 2.2 Hatcher Homology

Let $X$ be the 2 complex obtained from $S^1$ with its usual cell structure by attaching two 2 cells by maps of degrees 2 and 3 , respectively. (a) Compute the homology groups of all the subcomplexes $A\subset X$ and the corresponding quotient…
EgoKilla
  • 2,538
3
votes
2 answers

How to compute the fundamental group of this space?

I know that without the closed disk, a sphere with the a diameter deformation retracts onto the wedge sum of a circle and a sphere. But I can't figure out how to deform the disk to a suitable space.... Could anyone help me?
Keith
  • 7,673
3
votes
2 answers

Problem with understanding proof of Van Kampen's theorem

I'm currently reading J.P May's book, "A Concise Course in Algebraic Topology". I don't understand his proof of the fundamental groupoid version of Van Kampen's theorem, particularly the part where he proves that $\tilde\eta ([f])=\tilde\eta[g]$, if…
Ishan Banerjee
  • 6,459
3
votes
2 answers

How do I prove that the union of two simply connected open sets whose intersection is path connected is simply connected?

I'm trying to understand Ronnie Brown's answer here: union of two simply connected open , with open and non empty intersection in $R^2$ Let $X$ be a topological space and $U,V$ be simply connected open subsets of $X$ such that $U\cap V $ is…
Number 9
  • 1,457
3
votes
1 answer

Extending a homotopy between loops in a space to the entire space.

Let $X$ be a topological space, and let $f, g: S^1\to X$ be homotopic circles in $X$. Is there a continuous function from $X$ to itself that sends $f(S^1)$ to $g(S^1)$, and if so, how do I find it? This is related to a homework problem, so please…
Nishant
  • 9,155
3
votes
3 answers

unreduced suspension

Is the definition $SX=\frac{(X\times [a,b])}{(X\times\{a\}\cup X\times \{b\})}$ of the unreduced suspension the standard defininition? If I consider $X=$ point, the suspension of $X$ is a circle. But I saw an other definition of the unreduced…
user151465
3
votes
1 answer

Map into knot complement inducing quasi-isomorphism

I'm looking at the following topology qualifying exam question: Let K be a knot in $S^3$. Construct a map $f: S^2 \vee S^1 \rightarrow (\mathbb{R}^3 - K)$ that induces an isomorphism of integral homology. My biggest problem is I'm not really sure…
Joe
  • 1,310
3
votes
1 answer

Homotopy equivalence with a point need not be a deformation retract

This question arises from problem 8 on pg. 366 of Munkres. Let $X$ be the union of the sets $(1/n) \times I$, $0 \times I$, and $I \times 0$, where $I = [0,1]$, with the topology it inherits from $\mathbb R^2$. I am trying to prove that the…
user15464
  • 11,682
3
votes
0 answers

Homology of stunted infinite real projective space

Consider the following composite based map $$f: S^2 \xrightarrow{\sim} RP^2/RP^1 \to RP^\infty/RP^1$$ induced by the inclusion of the real projective plane $RP^2$ into infinite real projective space $RP^\infty$. Consider ordinary homology with…
user17982
3
votes
1 answer

Why are degree maps for cellular boundary formula from $S^{n-1}\to S^{n-1}$?

For a CW-complex, there's the cellular boundary formula that $$ d_n(e^n_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_\beta $$ where the coefficients $d_{\alpha\beta}$ are the degrees of the map $$ S^{n-1}\to X_{n-1}\to S^{n-1} $$ where the first map…
Josh Donaldson
  • 33
3
votes
1 answer

Why is $H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$ a split injection?

In Hatcher's Algebraic Topology, section 3.2, during the computation of the cohomology ring of a $n$-torus, the following assertion is made. Let $Y$ be a space and $R$ a commutative ring. Then the natural map in cohomology $$ H^n(I \times Y, R) \to…
Akhil Mathew
  • 31,310
3
votes
3 answers

More elementary proof that $\pi_n(S^n) \cong \mathbb{Z}$

The proof I know that $\pi_n(S^n) \cong \mathbb{Z}$ is based on the Hurewicz theorem (which implies that $\pi_n(S^n) \cong H_n(S^n)$). I'm looking for a more elementary argument - preferably something which doesn't require any homology theory. Any…
Paul Siegel
  • 9,117
3
votes
3 answers

The existence of a 1-1 continuous map between two topological spaces.

Show that there is no one-to-one continuous map $f$ from $\mathbb{R}^n$ to $\mathbb{R}^2$ for $n\gt 2$ with $f(0)=0$. I tried using the hint: consider $f:\mathbb{R}^n-\{0\}\rightarrow \mathbb{R}^2-\{0\}$ and the induced map: $f_*:\pi_1…
Toeplitz
  • 1,443
  • 10
  • 25
3
votes
1 answer

Hatcher 2.2 exercise 10

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim −x$ for $x$ in equator $S^1$. I want to compute the fundamental group and homology groups $H_i(X)$. I also want to repeat this exercise for $S^3$ with antipodal points of the…
Anton
  • 81
3
votes
1 answer

How does one triangulate the mapping cylinder of a diffeomorphism?

The question is fairly self-explanatory. In particular, I would like to know how to triangulate the mapping cylinder arising from applying a Dehn twist to the torus. The reason is that I am thinking of this mapping cylinder as a cobordism from the…
Gerrit Goosen
  • 41
1 2 3
…
99 100