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
0 answers

When is it true that $\widetilde{H}_n(\Sigma^k{\Omega^k{X}}) = \widetilde{H}_n(X)$?

where $\widetilde{H}$ denotes reduced homology, $\Sigma^k$ denotes suspension and $\Omega^k$ denotes looping. The context that I need this is to show that the homology groups of a prespectrum $T$ are the same as the homoloy groups of its left…
PhysicsMath
  • 1,151
3
votes
3 answers

Mapping from X to $S^4$

I found this question in a book (Topology II: homotopy and homology: classical manifolds) Show that the quotient space $X = S^2 \times S^2 / [(x_1,x_2) \sim (Rx_1,Rx_2)]$ where R is the reflection in the equatorial plane, is homeomorphic to $S^4$. I…
Srikanth
  • 689
3
votes
1 answer

Geometric construction of $J$-homomorphism

In D. Freed's notes eqn (5.32), he defines the $J$-homomorphism geometrically by considering the equatorial $n$-sphere as an $n$-submanifold of $S^m$, and giving it a framing that makes it null-bordant, then he claims that restricting to pointed…
PhysicsMath
  • 1,151
3
votes
0 answers

what is a (co)homology theory?

There are different axioms for a (co)homology theory: Homotopy invariance is always there, but for the rest there is or isn't: long exact sequence for pairs of topological spaces exact sequence for cofibrations additivity (for coproducts of…
user3267
  • 420
3
votes
2 answers

projective space, constant function, homotopy

Show, that every continious function $f:\mathbb{R}P^2\to S^1$ is a homotopy to a constant function. (Hint: Has $f$ a lift $\tilde{f}:\mathbb{R}P^2\to\mathbb{R}$) Hello, I want to solve this problem, but I am kinda stuck and the given hint just…
Mr.Topology
  • 362
3
votes
1 answer

Homotopy equivalence involving cell - complexes

I have two small questions related to some point-set topology involved in the following. My question is inspired after reading the proof of Proposition 1.26 of Hatcher. Suppose I have a space $X$ that is built out of a subspace $A$ that is…
user38268
3
votes
1 answer

How is it possible that $H_p(\Bbb{S}^n)\cong H_p(\Bbb{S}^{n-1})$?

My Algebraic Topology book says the following: For $\Bbb{S}^n$, $H_p(\Bbb{S}^n)=\Bbb{Z}$ for $p=\{0,n\}$, and $H_p(\Bbb{S}^n)=0$ otherwise. Also, by Mayer-Vietoris, $H_p(\Bbb{S}^n)\cong H_p(\Bbb{S}^{n-1})$. How can both be true? Shouldn't…
user67803
3
votes
0 answers

The universal property of pullback bundle

The universal property says the following: for any pair of maps $i: Z \to X$ and $j: Z \to E$ fitting into the following commuting diagram where $p: E \to B$ is a fiber bundle (by the way this is true for any Serre fibration but I am applying it to…
PhysicsMath
  • 1,151
3
votes
1 answer

Prove that the cone over $S^1$ is homeomorphic to $D^2$

As the title, I am thinking $$f :\ S^1\times I / S^1\times \lbrace1\rbrace \rightarrow D^2$$ as $f(x,y,z)= (x,y)$ and $$g :\ D^2 \rightarrow S^1\times I / S^1\times\lbrace1\rbrace$$ as $g(x,y)=(x,y,1-\sqrt{x^2+y^2})$. Just remains to show $f$ and…
Chrisc
  • 31
3
votes
0 answers

For which $g,p$ does $\Sigma_{g,p}$ cover $\Sigma_{3,2}$?

I am preparing for my qualifying exams. There is an algebraic topology problem I don't know how to do it. Thanks a lot for your help. Let $\Sigma_{g,p}$ denote the surface of genus $g$ with $p$ punctures. For what $g,p$ does $\Sigma_{g,p}$ cover…
user1773
  • 31
3
votes
1 answer

Attaching map of $(k+l)$-cell of $S^k \times S^l$ with its usual $CW$ structure.

Hatcher describes the Whitehead product as the composition of maps $S^{k+l-1} \to S^k \vee S^l \overset{f \vee g}{\to} X$. where the first map is the attaching map of the $(k+l)$-cell of $S^k \times S^l$ with its usual CW structure. I can't find a…
user7090
  • 5,453
  • 1
  • 22
  • 53
3
votes
0 answers

How are the group actions induced by the fundamental group and the group of Deck transformations different?

Let $\overline{X}$ be a universal covering space of $X$. If $p:\overline{X}\to X$ is the projection, then let $p^{-1}(x)$ be the fiber of $x\in X$, and let $q\in p^{-1}(x)$. Consider the action of the fundamental group $\pi_1(X)$ on $q$ in two…
user67803
3
votes
1 answer

$T^2$, the two dimensional torus

How is the two dimensional torus $T^2$, different than a torus? I'm supposed to construct a universal cover of $T^2$ as part of an assignment but I just want to make sure I'm working on the right problem. My guess is that there is no difference…
Bob
  • 1,462
3
votes
1 answer

Finding triangulations of spaces.

I am currently pursuing a course in basic homology theory and i am finding it really difficult to find the triangulation of spaces. I know that a triangulation of a topological space $X$ is a simplicial complex $K$, homeomorphic to $X$, together…
Abhishek Shrivastava
  • 854
3
votes
2 answers

Degree of maps on the sphere

Given a map $f:B^n \to S^n$, where $B^n$ is the unit ball and $S^n$ is the unit sphere, is it true that the degree of $f|_{S^n}$ is always 0, where $f_{S^n}$ is the restriction of $f$ to $S^n$? If so, why? Thanks!
user5352
1 2 3
…
99 100