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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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Homology of surface of genus $g$

This is a homework question given to me by someone of the community here and it's a generalisation of this. I was wondering if you could have a look and tell me if it's right. Thanks for your help! Task: Compute the homology of a surface of genus…
Rudy the Reindeer
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For every $k \in {\mathbb Z}$ construct a continuous map $f: S^n \to S^n$ with $\deg(f) = k$.

Suppose $S^n$ is an $n$-dimensional sphere. Definition of the degree of a map: Let $f:S^n \to S^n$ be a continuous map. Then $f$ induces a homomorphism $f_{*}:H_n(S^n) \to H_n(S^n)$ . Considering the fact that $H_n(S^n) = \mathbb {Z}$ , we see that…
Yoav Bar Sinai
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What does "splitting naturally" mean in the Universal Coefficients Theorem

The Universal Coefficients Theorem states that $0\rightarrow H_n(X)\otimes G\rightarrow H_n(X;G)\rightarrow\operatorname{Tor}(H_{n-1}(X),G)\rightarrow 0$ splits, but not naturally. In all the algebraic topology contexts I've come across, "natural"…
user9402
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3 answers

Star convex set is simply connected.

We define a set $S \subset \Bbb{R}^n$ to be star convex if there exists $a \in S$, such that the line segment connecting $a$ and any other point in $S$ lies entirely in $S$. I would like to show that it's simply connected. Can someone verify my…
Student1
  • 191
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Topological properties that aren't conserved over homotopy?

The problem is asking to list half a dozen topological properties that aren't preserved under Homotopy. I can only think of cardinality (contractible spaces), compactness($\Bbb R^n$ is contractible), and interval type (open vs closed both…
Oliver G
  • 4,792
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Why universal G-bundles are contractible?

Let $G$ be a nice topological group and $E\to B$ a universal $G$-bundle. I'm interested in a proof of contractibility of $E$ using only the universal property of it. I also know that if there is a contractible $G$-bundle, then all of others are also…
Mostafa
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In Algebraic Topology, why do we want to localize spaces?

I'm reading May's More Concise Algebraic Topology and the first half of the book seems to be written under the assumption that the reader has the motivation that we want to localize the underlying topological space when doing computations. Coming…
Naiche Cimarron Downey
  • 315
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3 answers

Is there a gap in the standard treatment of simplicial homology?

On MO, Daniel Moskovich has this to say about the Hauptvermutung: The Hauptvermutung is so obvious that it gets taken for granted everywhere, and most of us learn algebraic topology without ever noticing this huge gap in its foundations (of the…
Qiaochu Yuan
  • 419,620
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Why do we use the smash product in the category of based topological spaces?

I was telling someone about the smash product and he asked whether it was the categorical product in the category of based spaces and I immediately said yes, but after a moment we realized that that wasn't right. Rather, the categorical product of…
Aaron Mazel-Gee
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Ring structure in the Serre spectral sequence

I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table with the $E^2$-page of a spectral sequence…
dstt
  • 1,089
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Principal fibrations with a section are trivial

I'm trying to prove that, given a principal fibration $\Omega B \rightarrow F \stackrel{p}{\rightarrow} E$ such that $p$ is a retraction of $F$ onto $E$, the total space $F$ is homotopy equivalent to the product $\Omega B \times E$. This is…
Alex
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Is there a map from the torus to the genus 2 surface which is injective on homology?

Let $T$ denote the torus and $M_2$ the genus 2 surface. Specifically, I am wondering if there is a map $f\colon T\to M_2$ such that $f_*\colon H_1(T)\to H_1(M_2)$ is injective. By thinking about the explicit generators of each of these groups, it…
Ekie
  • 665
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3 answers

Smash and join products of spheres

How to prove rigorously that $\mathbb S^n \wedge \mathbb S^m =\mathbb S^{n+m}$ and $\mathbb S^n \ast \mathbb S^m = \mathbb S^{n+m+1}$? And what intuition should i have for compute $\wedge,\ast$ for difficult spaces?
qwenty
  • 1,540
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Is the Serre spectral sequence a special case of the Leray spectral sequence?

Let $F \to E \to B$ be a fibration with $B$ simply connected (more generally, such that $\pi_1(B)$ acts trivially on the homology of $F$). Then there is a Serre spectral sequence $H_p(B, H_q(F)) \to H_{p+q}(E)$. One can do the same for singular…
Akhil Mathew
  • 31,310
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The image of simply-connected domain

If $\Omega$ is a simply-connected domain in $\mathbb R^n$ and $f$ is a injective continuous map from $\Omega$ to $\mathbb R^n$, then is it necessary that $f(\Omega)$ a simply-connected domain?
Summer
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