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

Group bundle over a topological space

Suppose $p:\tilde X\rightarrow X$ is the universal cover of $X$. Take $G$ a group where $\pi_1(X,x)$ acts by isomorphisms. I read that if we consider $X\times G$ ($G$ with the discrete topology) and factor out the action of the fundamental group…
Bill
  • 952
3
votes
1 answer

Map between projective spaces induces trivial map on first homotopy groups

I have the following problem: Let $n>m>0$, show that every map $f:\mathbb{RP}^n\to\mathbb{RP}^m$ induces the trivial map on the fundamental groups. I paste the given solution below: Now, this solution looks wrong to me (or at least missing a lot…
Daniel Robert-Nicoud
  • 29,795
3
votes
0 answers

How to make the orbit space $T/G$ of torus $T$ homeomorphic to the Klein bottle?

Actually it is one of the exercises of Munkres. $G$ is a group of homeomorphisms of the torus having order $2$. How do I get $G$ in order to make $T/G$ homeomorphic to the Klein bottle? Can anybody give me a hint?
Keith
  • 7,673
3
votes
0 answers

Induced group action: homology vs cohomology

Let $M$ be a compact orientable manifold with finitely generated free abelian cohomology groups in even dimensions and $0$ otherwise. Conditions imposed on cohomology clearly imply that $H_i(M) \cong H^i(M)$ for any $i$ (throughout this post,…
a student
  • 31
3
votes
1 answer

Homology of 5-manifold

Let $M$ be a closed connected 5-manifold such that $\pi_1(M)=\mathbb{Z}_7$ and $H_2(M)=\mathbb{Z}^2$. I would like to compute its homology and cohomology. Standard computations involving Poincaré duality and the universal coefficient theorem for…
user54631
  • 707
3
votes
0 answers

Equivariance of a map from tom Diecks book

in the book "Transformation Groups" of tom Dieck on page 123 ff., an equivariant version of the Hopf classification theorem is developed. I extract the relevant data to state my question: (U, B) is a relatively free $n$-dimensional W-CW complex,…
DanielW
  • 283
3
votes
2 answers

Cell structure of $S^2 \times S^1$

Can anyone please provide the cell structure of $S^2 \times S^1$? I know that there are one cell in each dimension from 0 to 3 but I am not sure about the attaching maps. Thanks in advance.
Paladin
  • 1,053
3
votes
2 answers

A doubt in Hatcher's explanation of reparametrization.

Hatcher contains the following paragraph: Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. Reparametrizing a path preserves its homotopy class since…
freebird
  • 71
3
votes
1 answer

$\tilde{H}_i(S^n-X)$, $X$ a Finite Graph

I came across this question. Prove that $\tilde{H}_i(S^n-X)\cong H_{n-i-1}(X)$ if $X$ is a finite connected graph embedded in $S^n$. By Alexander Duality, this is true if the group on the right is a cohomology group instead of homology. I've been…
J126
  • 17,451
3
votes
0 answers

Coboundary map problem

I have to show that if $A$ and $B$ are compact connected subsets in the plane such that $A\cap B$ is not connected (and not empty), then $\Bbb R^2\setminus(A\cup B)$ is not connected. The tool I must use is the coboundary map. I set $U=\Bbb…
User
  • 1,173
3
votes
1 answer

An even map $S^n \to S^n$ of degree two

I started to wonder if there exist a continuous map $f \colon S^n \to S^n$ such that $\mathrm{deg} f = 2$ and $f$ is even, i.e. $f(x) = f(-x)$. This question is only interesting for odd $n$ since if $n$ is even then every even map $f$ has…
J. J.
  • 9,432
3
votes
1 answer

Deletion of disc

Let $M$ be an closed manifold, $D$ a disc inside it. As far as I understand, in orientable case the only difference between the homology (over a given field) of $M$ and $M \setminus D$ is one more dimension in $n-1$ homology: although Mayer-Vietoris…
evgeny
  • 3,781
3
votes
0 answers

Proof of Hurewicz' thm. in Hatcher

I'm having trouble understanding the last part of Hatcher's proof of Hurewicz' theorem. (It's on page 367, thm. 4.32). We want to show, that a cellular boundary map: $d:H_{n+1}(X^{n+1},X^n) \rightarrow H_n(X^n,X^{n-1})$ is a map $\oplus_\beta…
Kristoffer
  • 83
  • 5
3
votes
0 answers

What is the mapping class group of the wedge of circles?

I was wondering if there is a description of the mapping class group of a wedge of $n$ circles. Are the only kinds of homeomorphism classes in the mapping glass group are compositions of permutations of the circles and maps which send a circle to…
Felix Y.
  • 673
3
votes
0 answers

The long exact sequence of compact manifold

If $M$ is a compact orientable manifold with boundary, is it necessary that in the long exact sequence ${H_n}(M,\partial M)\mathop \to \limits^{{\partial _ * }} {H_{n - 1}}(\partial M)$, ${{\partial _ * }}$ sends $1$ to $+1$ or $-1$ in ${H_{n -…
Summer
  • 6,893
1 2 3
…
99 100