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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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Proof of excision

The following link is the proof of excision from Rotman's Algebraic Topology. Please see the the third line of the page 118. I am wondering why the equation is held. Note that it is composition of q Subdivisions here. Thanks very much.
Yuan
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A basic question on Relative Homology

And so this week, our algebraic topology class starts with relative homology groups. But there are some (REALLY) basic parts of the definition of the relative homology group that I don't understand why... Our class is currently using Hatcher's…
ireallydonknow
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Why is the fibre of each point compact?

For a compact covering space, the fibres of the covering map are finite. I am working on the same question as the one posed in this link, but there was an unanswered question at the end, namely, why is the fibre of EACH point compact. This just so…
Johnny Apple
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Relations between the homology group of quotient space and the relative homology group

Let $X$ topological space with subspace $A$.Under what conditions,$H_{n}(X,A)$ is isomorphic to $H_{n}(X/A)$
ABC
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Mapping cone z to z^2 on S1 is RP2

I want to show that the mapping cone of $z \mapsto z^2$ on $S^{1}$ is homeomorphic to $\mathbb{RP}^{2}$. My thought was: 1) $\mathbb{RP}^{2}$ is indeed $S^{2}$ with identifying each pair of antipodal points as one point. 2) We can assign "cross…
fiverules
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Developing a counter-example in algebraic topology

Two maps $f,g$ into $Y$ are n-homotopic if, for every complex $K$ of dimension at most $n$ and for every map $\phi$ of $K$ into $X$ the compositions $f \phi,g\phi:K \to Y$ are homotopic. As a sort of converse I am trying (it is an exercise in…
Juan S
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$H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1

Let $A,B$ be two open set of a topological space, $H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1, where the homomorphisms is induced by inclusion. I feel that using the Mayer-Vietoris exact sequence is the key to this question…
Kaa1el
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Cohomology Ring of Projective Space

I would like to see different ways of calculating the cohomology ring of $\mathbb{R}P^n$. I know there are several ways, for example, using Poincare Duality, Gysin sequence, etc... Sketch your favorite proof! PD. Is there a direct way (maybe a…
Manuel
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Universal covering of maximal torus in a compact Lie group

What is the universal covering of maximal torus $T$ of dimension $n$? of a compact Lie group $G$. Is there any reference?
user61135
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non homology based polynomial time computable invariants

I am wondering if there are polynomial time computable invariants of topological spaces (say, finite simplicial complexes, or finite CW complexes with computable attaching maps to be a bit more general) which are somehow fundamentally different from…
rhl
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Topology problem, $X=A\cup B$, show $\pi_1(X)\cong \pi_1(A)*\pi_1(B)$

$X$ is Hausdorff and locally arcwise connected, $X=A\cup B$, where $A$ and $B$ are closed subspaces of $X$, $A\cap B =\{ p\}$ and $p$ has open contractible neighborhoods $U,V$ in $A,B$ respectively. Show $\pi_1(X,p) \cong \pi_1(A,p)*\pi_1(B,p)$. I…
user43138
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Show that $A*B$ is 1-connected

When $A\subset\Bbb R^m,B\subset\Bbb R^n$ define $$A*B=\{((1-t)a,t,tb)\mid a\in A,t\in [0,1],b\in B\}\subset\Bbb R^{m+n+1}$$ If $A\not=\emptyset$ and $B$ is path-connected show that $A*B$ is $1$-connected It's enough to show $A*B$ is path-connected…
user63416
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Is the boundary of a 0-simplex empty?

This is mostly a question about conventions and is motivated by me trying to understand the definition of Delta-complex structure given in Hatcher. It is right to say that the boundary of 0-simplex is empty, or equivalently that a 0-simplex is…
GFR
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Exact meaning of homology

Possible Duplicate: Soft Question - Intuition of the meaning of homology groups I've been studying some homology recently and I know it supposedly counts $n$-dimensional holes. For example, the torus has first homology group $H_1(T^2) =…
Rudy the Reindeer
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Homology with local coefficients

Is there any relation between the homology of a space with local coefficients (in $\mathbb Q$ vector space) and the homology with coefficients in $\mathbb Q$? Thanks!
abc
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