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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
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Homology of punctured projective space

I am trying to calculate $H_k(X)$ where $X = \mathbb{R}P^n - \{ x_0 \}$ I started thinking about $k=2$. We can get the projective plane by taking the upper hemisphere with points along the equator identified according to the antipodal map. If we…
Juan S
  • 10,268
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$\mathbb{Z}$ vs. $\hat{\mathbb{Z}}$ coefficents in Singular Cohomology

Let $X$ be a finite CW complex, given singular cohomology with coefficents in $\hat{\mathbb{Z}}$, $H^i(X, \hat{\mathbb{Z}})$ can you recover singular cohomology with coefficents in $\mathbb{Z}$? I was told that one cannot do this. However, it seems…
user53932
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Is it possible for a closed manifold to deformation retract onto a proper subset of itself?

Let $M$ be a closed (compact, without boundary) topological manifold. Is it possible for there to exist a subset $A$ of $M$ such that $M$ deformation retracts onto $A$?
user15464
  • 11,682
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Torus as double cover of the Klein bottle

Reading through some lecture notes and it says The torus $T^2$ is the orientation double cover of the Klein bottle $K$, via the covering projection $p:T^2\to K; [x,y]\mapsto [x,2y]$ Could someone explain this map? Are they taking $[x,y]$ as the…
09867
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Finding the homology group of $H_n (X,A)$ when $A$ is a finite set of points

It is one of the problems in Hatcher's book. I need to find the homology group of $H_n (X,A)$ when $A$ is a finite set of points and $X$ is $S^2$ or $T^2$. I figured out that for $n>1$, I could use the long exact sequence and make $H_n (X,A)$…
Emily
  • 1,291
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$S^m * S^n \approx S^{m+n+1}$

I'm interested in showing that $S^m * S^n \approx S^{m+n+1} $, as discussed in exercise 0.18 of Hatcher's Algebraic Topology. One way to show it would be to show that $X * Y \approx \Sigma(X \wedge Y)$. This fact shows up only in a later exercise in…
Eric Auld
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Why is the complement of a discrete subspace of $\mathbb{R}^n$ ($n \ge 3$) simply-connected?

I'm stuck with an Exercise in Hatcher's Algebraic Topology. (Exercise 1.2.6) This problem asks me to show that the complement of a discrete subspace of $\mathbb{R}^n$ is simply-connected if $n\ge 3$, using the fact that if $Y$ is obtained by…
Wooyoung Chin
  • 647
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Fixed Points of a Reflection

This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows: Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that reflection across a plane $P$ determines a…
NHow
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Cellular boundary map for real projective space

I am trying to compute the cellular boundary map for real projective space: $\mathbb{R}P^n$. But, I am stuck. Any help will be appreciated. Here's what I have got so far. I want the map from $d: H_n(X^n,X^{n-1}) \to H_{n-1}(X^{n-1},X^{n-2})$. Now,…
doofus
  • 131
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What is the intended solution to exercise 0.16 in Hatcher? (Contractibility of $S^\infty$)

Hatcher, Algebraic Topology, Exercise 0.16 reads: Show that $S^\infty$ is contractible. Let's look at the definition Hatcher gives for $S^\infty$: There are natural inclusions $S^0 \subset S^1 \subset \cdots \subset S^n$, but these subspheres are…
Daniel McLaury
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Hatcher Problem 2.2.36

I am struggling with the following question (2.2.36) from Hatcher for quite some time now: Show that $H_i(X\times S^n) \simeq H_i(X) \oplus H_{i-n}(X)$. I don't know how to use the hint given by Hatcher. I have been trying to use the…
Dave
  • 131
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2 answers

Group action and covering spaces

Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition of covering space $X \rightarrow X/H \rightarrow…
ShinyaSakai
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Homotopy equivalence iff both spaces are deformation retracts

I'm trying to prove that $f: X \rightarrow Y$ is a homotopy equivalence $\iff$ $X$, $Y$ are both homeomorphic to a deformation retract of a space $Z$ The $\Leftarrow$ was not a problem. If both are deformation retracts it follows that $Z \simeq X$…
Rudy the Reindeer
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Visualising regular CW complex

I am somewhat struggling to see the difference between a regular CW complex and a non-regular CW complex. The difference is all the attaching maps are homeomorphisms - i.e. there are no identifications made on the boundary. So I guess if I produce a…
Juan S
  • 10,268
13
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1 answer

Fundamental group of $\mathbb{R}^3$ \ finite number of lines passing through origin.

I want to confirm my answer for this question: Calculate the fundamental group of $X=\mathbb{R}^3\setminus \{\text{union of n lines passing through origin}\}$. My idea is that $X$ deformation retracts onto $S^2\setminus \{\text{union of $2n$…
user51266
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