# 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 ...
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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 ...
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### Understanding Leray-Hirsch theorem from Bott and Tu.

What does the statement actually mean? Could anyone please share some ideas on it? Also how does it follow from Künneth formula? Any help in this regard would be warmly appreciated. Thanks for your ...
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