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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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Standard Decomposition of 3-sphere into two solid tori

I am working thru Hatcher on my own and having some problems with what he calls the "standard decomposition" of the 3-sphere into two solid tori. Specifically, (1) Why is the boundary of a 4-disk equal to the boundary of the product of two…
PossumP
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If X closed surface with $\pi_1$ infinite, does $\pi_2(X)=0$? (Hatcher exercise 4.2.16)

I'm trying Hatcher (Algebraic Topology) exercise 4.2.16. Show that closed surfaces with infinite fundamental group are $K(\pi,1)$'s by showing that their universal covers are contractible, via the Hurewicz theorem and results of section 3.3. So…
Ali
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Can we prove $H^1(X,\mathbb{Z})\cong Hom(\pi_1(X),\mathbb{Z})$ using torsors?

Let $X$ be a topological space, its first cohomology group $H^1(X,\mathbb{Z})$ classifies $\mathbb{Z}$-torsors over $X$. I think they are special kind of infinite sheet covering space of $X$. How can we show $H^1(X,\mathbb{Z})\cong…
user93417
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Homology of sphere-complements

I have to solve the following questions: "For a subset $X \subset S^n$ determine the homology group $H_i(S^n - X)$, where (a) $X \cong S^l \vee S^k$ (b) $X \cong S^l \sqcup S^k$ (disjoint union) " The second one I could solve by myself: I used de…
Donut
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Hatcher Corollary 4.12

Just to ask a quick question regarding a corollary 4.12 in Hatcher: "A CW pair $(X,A)$ is n-connected if all the cells in $X-A$ have dimension greater than $n$. In particularly the pair $(X,X^n)$ is n-connected, hence the inclusions…
yoyostein
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Calculate the Euler characteristic of a manifold equipped with a torus action

Suppose that $S^1$ acts freely on a manifold M. What is the Euler characteristic of $M$?
Peter Hu
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Motivations for Homology or Cohomology Theory

In some standard books on Algebraic Topology (except Hatcher's), the motivation for homology or cohomology theory is stated with the help of Cauchy's theorem, Green's theorem, Stoke's theorem etc (for example, one may see in books by Rotman, Massey,…
Groups
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Problem from Hatcher's book (3.3.25)

I have begun to read in Hatcher's book "Algebraic topology", about cohomology. In doing so, I have tried to solve some problems. I have difficulties with problem 3.3.25: Show that if a closed orientable manifold $M$ of dimension $2k$ has…
pascal
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How to prove that $T^1(M)$ is simply connected for some specific $M$.

Concretely, I'm working with the spaces: $S^n$, $\mathbb{C}P^n$ and $\mathbb{H}P^n$. I need to conclude that $T^1(M)$ is simply connected for all those manifolds $M$ I listed (with the exception of $S^2$). Now, I know that $T^1S^2 \cong SO(3) $,…
Aloizio Macedo
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Any manifold other than $SO(n\geq 3)$ having $\mathbb{Z}_2$ as the fundamental group?

It is hard for me to understand that $SO(3)$ has $\mathbb{Z}_2$ as the fundamental group. The problem is that, it is hard to image a mechanism such that there is a non-contractible loop which when repeated becomes contractible. So, is there any…
kaiser
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Spaces of Labeled Complexes (Munkres)

The following is taken from Munkres' Algebraic Topology book. I tried to determine which spaces (e.g. Mobius Strip, Klein bottle, etc) these complexes are, but to no avail. I computed the Euler Characteristics (V-E+F) to be -1, -1, 1, -1…
yoyostein
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Covering space calculation of figure eight.

I'm trying to do this calculation in Hatcher. So for the (1). I imagine cutting the loop at $a$ on the left call this Y and cutting the loop $a$ on the right call this Z. This will give you two subsets that you can apply Van Kampen to. The…
simplicity
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Continuous Map $f : S^n \rightarrow S^{n-1}$ with $f(-x) = -f(x)$

I'm studying for an algebraic topology exam, and the following question has me stumped. Problem. For $n \geq 1$, prove there does not exist a continuous map $f : S^n \rightarrow S^{n-1}$ such that $f(-x) = -f(x)$ for all $x \in S^n$. The case $n =…
Hans Parshall
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Difference between retract and deformation retract

I have a trouble with distinguishing retraction and deformation retraction intuitively. That is, deformation retraction is informally an operation on a space which continuously deform(for an example, expansion of a hole in a ball or compression…
Rubertos
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How can I conclude that F a continuous vector field on unit ball gives a map of degree zero?

It is a problem for a Hatcher's book, and it is my homework problem. It is a section 2.2 problem 3, stating: Let $f:S^n\to S^n$ be a map of degree zero. Show that there exist points $x,y \in S^n$ with $f(x)=x$ and $f(y)=-y$. Use this to show that if…
Emily
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