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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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Mapping cone of double suspension of Hopf map (Hatcher proposition 4L.11)

Let $\eta\colon S^3\to S^2$ denote the Hopf map. I don't understand a step in the proof of proposition 4L.11 of Hatcher, which states that $\eta\circ(\Sigma\eta)$ is not nullhomotopic. Let $C\eta=S^2\cup_\eta D^4$ the mapping cone of $\eta$. If we…
shin chan
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why $[f_r] =0$ for all $r$?

From Allen Hatcher Book Theorem $1.8$. Every nonconstant polynomial with coefficients in C has a root in $\mathbb{C}.$ In the proof of the theorem Hatcher say that as $r$ varies, $f_r$ is a homotopy of loops based at $1$. Since $f_0$ is the trivial…
jasmine
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$\mathbb Z_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even

Prop.2.29 on Hatcher's algebraic topology says $\mathbb Z_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even. If we have a unit sphere $S^2$ sitting in a standard coordinate, then let $f$ to be rotation of $S^2$ around $z$…
iefjkfdhfure
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Free action on CW complex induces cellular action

Suppose I have a topologycal space $X$, which admits a $CW-$structure, and a free $G-$action on $X$, where $G$ is a finite group. I am wondering whether I can choose a $CW-$structure on $X$ such that G acts freely on the cells of $X$. I know similar…
abho
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A Borsuk theorem

Let $M$ and $L$ be two subspaces of Banach space $X$ such that $\dim L<\dim M<\infty$. Let $S=\{m\in M : \|m\|=1\}$ and let $g$ be a continuous function from $S$ to $L$ such that $g(-m)=-g(m)$ for all $m\in S$. I need to prove that there exists an…
masmoudihoussem
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Understanding the boundary map in singular homology

I'm trying to understand the boundary map in singular homology: What does it mean for a $n$-chain to have no boundary? Here is my understanding: By definition of singular homology, any $n$-chain is just a finite sum $\Sigma_in_i\sigma_i$ where…
user388493
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fixed point of a continuous map on a projective space

Let $f:S^{2n} \rightarrow S^{2n}$ be a continuous map. Show that there exists $x \in S^{2n}$, such that $f(x) =x$ or $f(x) = -x$; any continous map $g: \mathbb R P^{2n} \rightarrow \mathbb RP^{2n}$ has a fixed point; I think 1 implies 2. So,…
sunkist
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Covering map with a right inverse is a homeomorphism

I am stuck on the following problem. I believe I am very close. Let $p:E\to B$ be a covering map. Suppose that there exists a continuous $f: B\to E$ so that $p\circ f = id_B$ (that is, $p$ has a right inverse). Prove that if $E$ is path connected,…
J-Q-McNeedHelpson
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Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z| \leq 2\}$

Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z| \leq 2\}$. I would like if in the proof the tools of algebraic topology were preferred over the other tools of analysis, complex analysis, algebra etc.
Dávid Natingga
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Hatcher exercise 1.1.18

Using Lemma 1.15, show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n\geq 2$, then the inclusion $A\hookrightarrow X$ induces a surjection on $\pi_1$. Proof. Note that $X = A\sqcup_{x\sim…
one potato two potato
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What happens if we remove requirement for cofinite vanishing in simplical chains

Let $\mathcal{K}$ be a simplical complex. Define (simplical) $n$-chains as maps from $n$-simplices in $\mathcal{K}$ to $\mathbb{Z}$ such that they vanish cofinitely many times. We then get a basis for a free abelian group "for free" since every…
locally_convex
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show that $\Sigma(S^1 \times S^1) \simeq S^2 \vee S^2 \vee S^3 $

Hatcher in Chapter 4 gives the example: $\Sigma(S^m \times S^n) \simeq S^{m+1} \vee S^{n+1} \vee S^{m+n+1} $ relating the suspension and the wedge product. for example $S^1 \vee S^1 $ looks like a figure eight. $\Sigma(S^1 \times S^1) \simeq S^2…
cactus314
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CW Structure for a torus with $g-$holes

How can I determine the CW Structure of a torus with $g-$holes? I understood that it consists of $4g$ $0-$ cells and $4g$ $1-$ cells. I understood well how can I form the skeletons $X_0$ and $X_1$ but I am unable to find maps $\phi : D^2 \to X $…
zero2infinity
  • 753
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Proof of convex spaces are homotopy equivalent to a point

In order to prove that a convex subpace $X \subseteq \mathbb{R^n}$ is homotopy equivalent to a point I did the following: Let $X \subseteq \mathbb{R^n}$ be a convex subspace in Euclidean space and $Y=\{0\}$ be a one point space. Let $x \in X$ and…
mathripper
  • 739
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Classificant map real projective space.

Let $\tau:=(T\mathbb{R}P^n, \pi, \mathbb{R}P^n)$ be the tangent bundle of $\mathbb{R}P^n$. The group $GL(n,\mathbb{R})$ acts on $\pi^{-1}(x) \simeq \mathbb{R}^n$ so we can consider $T\mathbb{R}P^n$ as a $GL(n)$-principal bundle and we can reduce the…
ArthurStuart
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