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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.

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21356 questions
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Cohomology of Hawaiian earring?

Do the infinite wedge of circles and the Hawaiian earring have the same cohomology? I am happy that they have different homologies (the first is countably generated, the second uncountably).
Jeremy Judge
  • 211
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Compute $\pi^n(S^1\times S^{n+1})$.

What is the space of homotopy classes of maps $S^1\times S^{n+1}\to S^n$? Is there a simple way to compute it, if we know $[S^{n+1}, S^n]\simeq\mathbb{Z}^2$ (resp. $\mathbb{Z}$ for $n=2$)?
Peter Franek
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Determining the induced map on homology $\tilde{H}_n(\mathbb{R}^n-\{0\})$ of $f\colon \mathbb{R}^n\to\mathbb{R}^n$ based on sign of $\det(f)$.

I'm having difficulty understanding the following. It appears as Exercise 7, p. 155 in Hatcher's Algebraic Topology: (this is not homework, by the way) For an invertible linear transformation $f\colon\mathbb{R}^n\to\mathbb{R}^n$ show that the…
Noomi Holloway
  • 1,682
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Maps of Wedges being Nullhomotopic

In Spanier's "Infinite Symmetric Products, Function Spaces and Duality," he makes the following claim: Given some $X\hookrightarrow S^n$, and $X'$ which is an "$n$-dual" of $X$ (i.e. for some $k$, and all larger $k'$,…
Jonathan Beardsley
  • 4,503
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How to compute the fundamental group of $S^2 / A$, where $A$ is a finite set of points?

Along the way to a much simpler solution to a homology problem, I thought about computing the fundamental group of $S^2 / A$. I quickly ran into trouble, so I want to know if there is there a slick way to do this. (For nontrivial |A|, of course.) I…
Elle Najt
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Comparing the torus and a small wedge product, and their universal coverings

I have the following problem: Let $A = S^1 \times S^1$ and $B = S^1 \vee S^1 \vee S^2$. Compute their universal coverings. Prove that $A$ and $B$ have isomorphic homology groups for any $n \in \mathbb{Z}_{\geq 0}$, but that their universal…
Bachmaninoff
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Local degree of a map

Suppose $f: X \to Y$ is a homeomorphism and $X,Y$ are orientable manifolds. For $U \subset X$ and $V \subset Y$ both open balls homeomorphic to $\mathbb{R}^n,$ we can define the local orientation at a point $p \in U$ as the relative homology group…
user142700
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CW construction of a generalized lens space from Hatcher

I am having trouble understanding one step in Hatcher's construction of generalized lens spaces (p 144 of Hatcher's Algebraic Topology For the generalized lens space $L_m(\ell_1,\ldots,\ell_n)$ he shows by induction that he can build Lens spaces…
Atticus Christensen
  • 1,509
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The coeffcients of a generator of $H_0(X)$ sum to $\pm 1$?

I'm reading Theorem 4.14 (p. 70) of Rotman's Intro to Algebraic Topology. He proves that if $X$ is a nonempty path connected space, then $H_0(X)\simeq\mathbb{Z}$, and if $x_0,x_1\in X$, then $cls(x_0)=cls(x_1)$ is a generator of $H_0(X)$. Here…
HD.
  • 41
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Homotopy equivalence gives an outer automorphism of the fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f \colon X \to X$ gives a well-defined element $g \in \operatorname{Out}(G)$. How might one begin on this question?
victor
  • 71
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On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?

I'm confused about part of the proof of Proposition 2.7 in Hatcher. If $X$ is nonempty and path-connected, then $H_0(X)\simeq\mathbb{Z}$. Hence for any space $X$, $H_0(X)$ is a direct sum of $\mathbb{Z}$s, one for each path-component of…
Hailie Mathieson
  • 1,950
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Is the suspension of a $\pi_n$ isomorphism a $\pi_{n+1}$ isomorphism?

Let $$f_*:\pi_n(A)\to\pi_n(X)$$ be an isomorphism induced by a map $f:A\to X$ of based topological spaces. Does the suspension $\Sigma f$ induce an isomorphism $$(\Sigma_f)_*:\pi_{n+1}(\Sigma A)\to\pi_{n+1}(\Sigma X)?$$
John Zhao
  • 411
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$S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $

Consider $S^m$ embedded in $S^n$ $ (m < n ) $ as the subspace $ \left\{ (x_1, x_2, ..., x_{m+1}, 0,...0) | \sum x_i^2 = 1 \right\} $. Show that $ S^n \backslash S^m $ is homotopy equivalent to $ S^{n-m-1} $ My thoughts: I've first considered the…
TRY
  • 397
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When do homology groups have torsion?

Let $C_q(K)$ be a group of $q$-chains on a given simplicial complex $K$. Since this is a free abelian group, its subgroup must be a free abelian group, especially $Z_q(K),B_q(K)$. Then we define the homology group $H_q(K)$ as $Z_q(K)/B_q(K)$. And…
Mike
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3-fold connected covers of punctured torus

I am interested in finding all the connected surfaces (up to homeomorphism) that can be described as $3$-fold covering spaces of the torus with a disc deleted (in both cases that the disc is closed or open). 1)How do we treat the "closed disc…
Bernard
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