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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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Covering Spaces for $S^1 \vee S^1$

I'm trying to list the connected 3-sheeted covering spaces of $S^1 \vee S^1$ up to isomorphism. I know what the answer is; I've seen listings. My question is, if I'm deriving this, how do I know I've got all the covers? Is there a nice…
ec92
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CW complex of a product

Another one that has been bugging me: Say $X$ is a finite CW complex. What is the simplest CW structure on $S^n \times X$ So I assume that $E$ is the family of cells in $X$ and $\Phi = \{ \Phi_e:e \in E \}$ is the family of attaching maps…
Juan S
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Euler characteristic of this polyhedron?

I´m trying to obtain the Euler characteristic of this polyhedron $P$, that is homeomorphic to the torus $T$ (I think): So it should be $\mathcal{X}(P)=\mathcal{X}(T)=0$. But we get $V=16, F=10, E=24$, so $\mathcal{X}(P)=2$. However, if we consider…
LH8
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Simplicial homology and homeomorphisms

In Hatcher's book, in the introduction page of singular homology, he mentions that "it is obvious that homeomorphic spaces have the same singular homology, in contrast to simplicial homology". However I thought that this was also true for simplicial…
NDewolf
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$X$ is contractible if and only if there exists a deformation retract to a point

We say that $X$ is contractible if it is homotopy equivalent to a singleton set. Let $A \subseteq X$. $A$ is a retract of $X$ if there exists a continuous map $r:X \rightarrow A$ such that $r(a) = a \; \forall a \in A$. A retract $A$ is a…
Matt
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how to compute the homology groups of a torus with n points removed?

i have really big problems in computing the homology groups and even don't really know how to write the exact sequence for different spaces. i tried to take the torus with n points removed,homotopic to a circle with n circles attached around and use…
msionl
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Natural examples of deformation retracts that are not strong deformation retracts

The question here asks for examples of deformation retracts which are not strong deformation retracts. A comment is given refering the asker to Exercise 0.6 in Hatcher. (The description of this space also appears in this question, which is about…
Dejan Govc
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Covering spaces and cohomology

If $p: B \rightarrow C$ is a finite covering space with covering group $G$. Why (rigorously) $H^{*}(B)^{W} \simeq H^{*}(C)$
ArthurStuart
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About the covering spaces of comb space.

I am working in this exercise: Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the four sides of the square $[0,1] \times [0,1]$ together with the segments of the vertical lines $x = 1/2, 1/3, 1/4,\dots$ inside the square. Show that …
Nell
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Why don't all path-connected space deformation retract onto a point?

I'm reading Hatcher's book Algebraic Topology and on page 12 he writes: ... It is less trivial to show that there are path-connected spaces that do not deformation retract onto a point... The definition of deformation retract is given as: A…
Rudy the Reindeer
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Hatcher pg 187: Idea of Cohomology

I am readying Hatcher page 178 where he tries to give the idea of cohomology in the case where our space $X$ is a graph. Now in the 4th paragraph from the top he says this: The cohomology group $H^1(X,G) = \Delta^1(X;G)/\textrm{Im} \delta$ will be…
user38268
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homotopic between two maps imply the homotopy between their mapping cone

Recall the mapping cone of a map $f: X\rightarrow Y$ is defined as the space $C_f: X\times [0, 1]\dot{\cup} Y/\sim$, where $\sim$ is the equivalence relation given by $(x, 1)\sim f(x)$ and $(x, 0)\sim(x', 0) $ for all $x, x'\in X$. Show that if $f,…
ougao
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Connectivity of a map and the long exact sequence of homotopy groups

Let $f:X\to Y$ be a map between connected CW complexes and $k\geq 0$ an integer. I am confused by the definition of $k$-connectivity or more fundamentally by what induces a long exact sequence of homotopy groups. My favourite definition for…
Daniel Dreiberg
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Homeomorphisms on a finite connected graph $X$ with $H_1(X; \Bbb Z)$ free abelian

For context, this is exercise 2.2.42 in Hatcher's Algebraic Topology. Let $X$ be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that $H_1(X; \Bbb Z)$ is free abelian of rank $n>1$, so the group of…
G Pace
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Regarding fundamental group computation of the complement of a torus knot in $S^3$

This is regarding the fundamental group computation of the complement of a toral knot in $S^3$ in Hatcher's algebraic topology book. See page 48. I have understood till the stage where the cross section of the torus minus the knot…
Karthik C
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