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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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Why is the one-point compactification of $S^5\times \mathbb{R}$ not a manifold?

Why is the one-point compactification of $S^5\times\mathbb{R}$ not a manifold? This problem was on qualifying exam, but I could not prove it. By simple drawings, one can see that the one-point compactification of $S^0\times \mathbb{R}$ and…
Rubertos
  • 12,491
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Finding homotopy equivalence

This is part of a problem from Hatcher: Show that the space in $\mathbb R^2$ which is the union (for $n \in \mathbb N$) of circles $C_n$, where $C_n$ is the circle centered at $(n,0)$ with radius $n$ is not homoemorphic to the wedge sum of infinite…
Sara
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Fundamental group of a knot

If the circle and any knot are homeomorphic as topological spaces, why do they have different fundamental groups?
April
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Why we need Cup product on different topological spaces.

I know the meaning of cohomology groups. These groups give some information about the topology of the space like connectedness. Also these groups give information about the different dimensional holes in topological spaces. I want to know that way…
King Khan
  • 1,046
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Does there exist a covering map from $p\colon X \to S_1 \vee S_1$

Covering map from $p\colon X \to S_1 \vee S_1$. X= I know infinite graphs with four edges incident at each vertex can be 2-oriented, but I don't think this would help me. I also need to find out $\pi_1(X)$ afterwards. So I can't use this in the…
Bob
  • 421
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Every topological abelian monoid $G$ is a product of Eilenberg-Mac Lane spaces

That's a classical theorem: I last stumbled upon it in this blog post, for example. But I don't know how to prove it. Can anyone provide a complete proof, or a reference? Here are some references that do not contain proofs. Aguilar-Gitler-Prieto,…
Bruno Stonek
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2 answers

Two Queries on Proof of Borsuk-Ulam in Bredon's $\textit{Geometry and Topology}$

I am reading through the proof of Borsuk - Ulam in Bredon, I have appended the proof below for reference. There are two things I don't understand in this proof: 1) In the first picture below, he writes " For any simplex $\sigma : \Delta_p \to X$,…
user38268
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Is there an embedding of the complex projective plane in complex space?

I've seen that the real projective plane $\mathbb{RP}^2$ can be embedded in $\mathbb{R}^4$. (see Wikipedia for example) Can the complex projective plane $\mathbb{CP}^2$ be embedded in $\mathbb{C}^4$. If so, what is the embedding?
zooby
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Fundamental group of the plane with infinite puncture

How to calculate the Fundamental Group of $\mathbb{R}^2 \setminus \mathbb{Z}^2$. I know that the Fundamental group of the Plane with $k$ punctures is the free product of $k$ copies of $\mathbb{Z}$. My guess is that $\pi (\mathbb{R}^2 \setminus…
Walker
  • 51
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Inclusion induces isomorphism in relative homology

I want to prove that inclusion $i:(B^n,S^{n-1})\to (B^n,B^n\setminus\{0\})$ induces isomorphism in relative homology. Now, $i$ obviously induces isomorphism of $H_p(B^n)$ with itself for every $p$. Restriction of $i$ to $S^{n-1}$ induces isomorphism…
Madara Uchiha
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4
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The fundamental group of some space

I have been solving some past exam questions and I came across the following question. Let $X=R^3-${(x-axis)U (y-axis)} be the complement of the x and y-axes in $R^3$. Compute the fundamental group of $X$. I am not really sure how to go about this.…
smanoos
  • 3,061
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Show that every continuous map $f:M\longrightarrow M$ which is homotopic to the identity has a fixed point

Let $M$ be a 4-dimensional closed simply connected manifold. Show that every continuous map $f:M\longrightarrow M$ which is homotopic to the identity has a fixed point. We have the Lefschetz Fixed Point Theorem that says: If $X$ is a finite…
Michael
  • 339
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A non-null-homotopic map from $T^{3}$ into $S^{2}$

Suppose we collapse the 2-skeleton of $T^{3}$ and we denote this map by $q$, \begin{equation} q:T^{3}\rightarrow S^{3} \end{equation} Then compositing $q$ with Hopf fibration, \begin{equation} S^{1}\hookrightarrow S^{3}\rightarrow…
user354914
  • 51
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Prove that for a path connected topological group $G$ the Euler characteristic $\chi (G)$ is zero.

Prove that for a path connected topological group $G$ the Euler characteristic $\chi (G)$ is zero. I'm trying to use Lefschetz for this proof. Bare with me. The Lefschetz Fixed Point Theorem states that: If $X$ is a finite simplical complex, or…
Michael
  • 339
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2 answers

Topology of complex projective plane

It is well known there are two ways to construct topology of $\mathbb{C}P^n$: quotient space of $S^{2n+1}$ by identifying $x$ with $\lambda x$, where $\lambda$ is complex nonzero constant. According to cell complex, quotient space of $D^{2n}$ by…
user301120
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