# 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.

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### Homology of the complement of a simple closed curve in a surface

Let $S$ be a closed orientable surface of genus $g$. Let $C\subset S$ be a simple closed curve. What is the homology of $S - C$? When I try to apply Mayer-Vietoris to $(S, S-C, T)$, where $T$ is a ...
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### What do the elements of the chains of a simplicial complex represent?

I've just started to learn homology and I don't quite understand why we define chains the way we do. For a simplicial complex $S$ we define $C_k$ to be the $k$-chains on $S$ given by an abelian group ...
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### Simply connected imply any two paths are freely homotopic

I know that simply connected ($\pi_0(p)=0$ and path-connected) is equivalent to "the space is path-connected and any two paths with the same endpoints are homotopic". What about two paths ...
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### Motivation behind fundamental group. [duplicate]

I am a graduate student.I am currently studying algebraic topology.Many things in algebraic topology are not clear to me.For example,the study of fundamental groups.I am unable to understand why we ...
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### uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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### For $U$ open subset of $\mathbb{R}^n$, assume $U=U_1\bigcup ... \bigcup U_m$ where all $U_i$ open and subconvex. Prove $H^j(U)=0$ for $j\geq m$..

A subset is subconvex if it is either convex or empty (so that the intersection of any two subconvex sets is subconvex). Note, $H^*(U)$ denotes the cohomology ring of $U$. The proof for this can be ...
1 vote
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### Suspension of a homology 3 sphere

Let $M^3$ be a homology sphere: a connected closed 3-manifold with the same homology groups as $S^3$. Calculate the first fundamental group and homology groups of the suspension $\Sigma M$. Use this ...
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### is open set in subspace equal to intersection open set in whole space with same subspace? [closed]

Is open set in subspace equal to intersection open set in whole space with same subspace?
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### The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$

I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. ...
1 vote
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### Lack of detail in Hatcher's proof of characterisation of the HEP

In page 14 of Algebraic Topology, Hatcher sketches a proof for the fact that a pair $(X, A)$ has the HEP iff $(X\times \{0\})\cup (A\times I)$ is a retract of $X\times I$. However, the argument rests ...
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### Prove there is no covering spaces

Prove that there are no covering spaces $p: S^{1} \longrightarrow C$ whose base is a convex space and whose total space is $S^{1}$ Covering space def: Let $B$ be a space. A map $p:E\longrightarrow B$ ...
1 vote
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### Example of connectivity of join is strictly greater than the sum of connectivities plus 2?

Given a topological space $X$, $X$ is $k$-connected if any $-1\le \ell \le k$ and continuous map $f:S^{\ell}\to X$ can be extended to $\bar{f}:B^{\ell+1}\to X$, where $S^\ell$ is viewed as the ...
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### Covering the Circle with closed connected sets.

This is a follow up to this previous post . The answer and the question got me to thinking about the case of $S^{1}$. The conditions are again the same . Can $S^{1}$ be covered by finite number of ...
1 vote