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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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Aspherical manifold with abelian fundamental group

Let $ M $ be an aspherical closed manifold with an abelian fundamental group. The fundamental group of an aspherical manifold is torsion-free and the fundamental group of a closed manifold is finitely generated. So $$ \pi_1(M) \cong…
Ian Gershon Teixeira
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Antipodal map of spheres

Let $A: S^2 \to S^2$ be the antipodal map between spheres, we know that it induces the endomorphism $-1$ on the $\pi_2(S^2)=Z$. Consider the composite $S^2 \xrightarrow{A} S^2 \xrightarrow{\pi} \mathbb{R}P^2$, the second arrow being the quotient…
gregodom
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How does one prove that any self homeomorphism of $\mathbb{C}P^2$ is orientation preserving.

I think this question is not very difficult,but I don't solve it.
karlbmg
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Hatcher 2.1.18: Calculating $H_{1}(\mathbb{R}, \mathbb{Q})$

I am trying to solve this exercise and wanted to check my approach. Question: Show $H_{1}(\mathbb{R}, \mathbb{Q})$ is free abelian and find a basis We get the LES from pairs $H_{1}(\mathbb{R}) \longrightarrow H_{1}(\mathbb{R}, \mathbb{Q})…
thorwi
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Questions about chain complex whose coefficient is the $\mathbb{Z}[\pi_1(X)]$-moudle

Let $R[G]$ be a group ring. Then, I have two questions about the chain complex, $C_n(\overset{\sim}{X}; \mathbb{Z}[\pi_1(X)] )$ (where$R=\mathbb{Z}, G=\pi_1(X)$, and $\overset{\sim}{X}$ is a universal cover of $X$). For example, let $M=S^1$ and…
Hanwoong Cho
  • 885
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The acyclic carrier theorem

In Mosher & Tangora's book Cohomology Operations and Operations in Homotopy Theory, they define an $h$-equivariant carrier as in the excerpt below. But I don't quite understand the definitions here. $\pi$ is a group,$K$ is a $\pi$-free chain complex…
Flying pencil
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If $M$ is a compact connected $4$-manifold such that $H_2(M;\mathbb{F}_2) \neq 0$, then the cup product map on $2$-nd degree cohomology is surjective.

Let $M$ be a compact connected $4$-manifold such that $H_2(M;\mathbb{F}_2)$ is non-zero. Show that the cup product map $H^2(M;\mathbb{F}_2) \times H^2(M;\mathbb{F}_2) \to H^4(M;\mathbb{F}_2)$ given by $(x, x) \mapsto x \cup x$ is surjective. Now…
Perturbative
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If $X$ is a connected $n$-dim manifold and $Y \subseteq X$ a proper subspace. Prove that $H_n(Y;\mathbb{Z}) \to H_n(X;\mathbb{Z})$ is the zero map.

If $X$ is a connected $n$-dimensional manifold and $Y \subseteq X$ a proper subspace. Prove that $H_n(Y;\mathbb{Z}) \to H_n(X;\mathbb{Z})$ is the zero map. My idea was to look at the long exact sequence in homology and use exactness properties to…
Perturbative
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Let $h: S^4 \to S^4$ be a homeomorphism, and let $f: S^4 \to S^4$ be a continuous map that is homotopic to h. Prove that $h(x)=f(x)$ has a solution.

Here's what I've done so far (correct me if I'm wrong). Showing that $f(x) = h(x)$ has a solution is equivalent to showing $h^{-1}(f(x)) = x$ has a solution since $h$ is a homeomorphism. Since $f$ is homotopic to $h$, by composing with $h^{-1}$, we…
fosterc4
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Intuition for Homology group

I am reading homology from Hatcher but I am not getting what's homology group? I am calculating homology group of different spaces , but it's not clear to me what am I really doing? Why do one consider the singular n simplices and then by taking the…
SOUL
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Brouwer degree and homotopy invariance

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $f \in C^1(\overline{\Omega})$. Following Topological degree theory and applications, we define the degree of $f$ with respect to some regular point $p \notin f(\partial \Omega)$ by…
Seirios
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Technical lemma to construct long exact sequence in homotopy

Let's define : $PY:= \left\lbrace \gamma : I \longmapsto Y : \gamma(0) = y_0 \right\rbrace$ given $f : X \longmapsto Y$ the homotopic fiber $F(f) := \left\lbrace (x,\gamma) \in X \times PY : f(x) = \gamma(1) \right\rbrace$. The space loop on $X$…
jacopoburelli
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On the Hopf invariant

It was an important problem of topology to determine for which dimensions the Hopf invariant was one. There are several clear expositions giving the definition of the Hopf invariant including the Wikipedia article in the link of this post. However…
snailspace
  • 857
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Quotients of infinite dimensional sphere

Recall $$S^\infty = \cup S^n,\ {\bf RP}^\infty = \cup {\bf RP}^n,\ {\bf CP}^\infty = \cup {\bf CP}^n$$ Hence $S^\infty / {\bf Z}_2 ={\bf RP}^\infty$. And I think that the following is possible : ${\bf CP}^\infty = \cup S^{2n+1}/S^1=S^\infty…
HK Lee
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Simplicial homology computation over $\mathbb{Q}$

I am learning about simplicial homology from Hatcher's Algebraic Topology book. He has examples of some simplicial homology computations for familiar/simple spaces (the torus, circle, and $\mathbb{R} \mathbb{P} ^2$). However, I am interested in…
Compact
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