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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Quantization of Chern number $c_1^n$ on 2n dimensional spin manifold

All orientable 2-manifolds are spin manifolds, and we know that the quantization of the first Chern number $c_1$ of a complex line bundle on 2-manifold is $\mathbb{Z}$. For 4-manifolds, the second Chern number $c_1^2$ of a complex line bundle is…
Chao-Ming Jian
  • 215
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For what manifold is boundary given odd-dimensional projective space?

Take projective real space $\mathbb P_n (\mathbb R)$ of ODD dimension. It is easy to proof that all his Stiefel-Whitney numbers are zero . So according Thom theorem there must exists manifold $M$ with boundary such that boundary is $\partial M=…
evgeniamerkulova
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Null-homotopic Maps from $S^n$ to $S^1$ for $n \gt 1$.

I'm not sure how to answer this one. Is every continuous map $f:S^2 \to S^1$ null-homotopic? If $n > 1$, where $n$ is a natural number, is every continuous map $f:S^n \to S^1$ null-homotopic?
james
  • 115
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Universal Cover of wedge sums of spaces?

I was wondering if there is some general prescription for picturing the universal cover of $X \vee Y$ for nice enough $X$ and $Y$. (Say, path-connected, locally path-connected and semilocally simply connected). In the case where one space is simply…
Nick Bhattacharya
  • 101
7
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Question about Surface of genus g

I understand the construction of a torus from a square by pasting opposite edges of a square and also its CW structure of that. It's easy to imagine. But how to understand the construction of a surface of genus 2g from a polygon with 4g sides and…
Kannan
  • 91
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A (somewhat) conceptual proof that the boundary of a fundamental class of a manifold with boundary goes to a fundamental class?

In this set-up, let M be a compact n-dimensional manifold with boundary $\partial M \neq \emptyset$. Assume that M is orientable, and that $[M] \in H_n(M,\partial M;R)$ is the fundamental class of M. If you haven't seen orientability / fundamental…
user101036
  • 818
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$X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ is null-homotopic

I want to show the following: $X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map For this I think I can use the following: $X$ $n$-connected $\iff $ every…
Rudy the Reindeer
  • 46,359
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$H_1(\mathbb{R}, \mathbb{Q})$ is free abelian

I'm trying to show that $H_1(\mathbb{R}, \mathbb{Q})$ is free abelian, this is another exercise in Hatcher. I'm not sure but I thought I can use the exact sequence $$ \cdots 0 \xrightarrow{f} H_1(\mathbb{R}, \mathbb{Q}) \xrightarrow{g}…
Rudy the Reindeer
  • 46,359
7
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2 answers

Does every continuous map $f : \mathbb{R} \mathbb{P}^n \to \mathbb{R} \mathbb{P}^n$ have a fixed point for $n = 2$? $n = 3$? and $n = 4$?

I am trying to solve this question but I don't have any idea on how to start. Thanks! Does every continuous map $f : \mathbb{R} \mathbb{P}^n \to \mathbb{R} \mathbb{P}^n$ have a fixed point for $n = 2$? $n = 3$? and $n = 4$?
Tom
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Homology relative to a point

I want to prove $$ \widetilde{ H_n}(X)\cong H_n(X, \ast)$$ The long exact sequence for pairs gives me $H_n(X) = \widetilde{H_n}(X) \cong H_n(X, \ast)$ for $n>0$ and $\widetilde{H_0}(X) \cong \widetilde{H_0}(X, \ast) $. So I would like to show…
Rudy the Reindeer
  • 46,359
7
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Excision in homology: $H(D^2, S^1)$

I've been trying to find an example of a not too obscure space for which one needs the excision theorem to compute the homology groups: Excision: If $Z \subset A \subset X$ where $A, U$ are subspaces of $X$ and $U$ is a subspace of $A$ then if…
Rudy the Reindeer
  • 46,359
7
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2 answers

Homology of $\mathbb{R}^3 - S^1$

I've been looking for a space on the internet for which I cannot write down the homology groups off the top of my head so I came across this: Compute the homology of $X : = \mathbb{R}^3 - S^1$. I thought that if I stretch $S^1$ to make it very large…
Rudy the Reindeer
  • 46,359
7
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$S^{2n}$ is the universal cover of $B$, what is $\pi_1(B)$.

Some students and I have tried to solve this problem in the following ways: Using degree theory and results about deck transformations. Using that $S^{2n}$ is the covering space of $\mathbf{RP}^{2n}$ and trying to see if $\mathbf{RP}^{2n}$ must…
john w.
  • 653
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1 answer

Linking Number and Intersection Number

Let $\Sigma$ be a smooth compact surface in $\mathbb{R}^3$ (for simplicity). If a closed curve $\gamma$ is linked with $\partial \Sigma$ with linking number $n$ (mod 2) then $\gamma$ should (generically) intersect $\Sigma$ in $n$ points (mod 2). …
GaryN
  • 73
7
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Topological proof to Hatcher's exercise 2.1.14

The first part of the exercise goes like this: Determine whether there exists a short exact sequence $0 \rightarrow \mathbb{Z}_4 \rightarrow \mathbb{Z}_8 \oplus \mathbb{Z}_2 \rightarrow \mathbb{Z}_4 \rightarrow 0$. It turns out the answer is yes,…
Skalf
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