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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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Monodromy groups and the choice of a base point

For a path connected topological space $X$, the fundamental groups at two different base points are isomorphic. Does the same hold for the monodromy group of a covering map $Y \rightarrow X$, when $X$ is path connected?
jonny
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How to prove the commutative diagram

Let $p_2:\mathbb S^1 \to \mathbb S^1$ be the two-sheeted covering map $p(z)=z^2$.If $f$ is odd($f(-z)=-f(z)$),show that there exists a continuous map $g:\mathbb S^1 \to \mathbb S^1$ such that $\deg f=\deg g$ and the following diagram commutes:$p_2…
Proton
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$2$ and $3-$fold covers for the figure $8$ graph

I have an assignment where I have to list all $2$ and $3-$fold covers for the figure $8$ graph and I am having some difficulties on how to think about it. As far as the degree $2$ are concerned, first. Is there a way of intuitively picture the…
user194469
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Find explicit cylcles representing generators of the infinite cyclic groups $H_n(D^n,\partial D^n)$

Here are some questions arising when I read Hatcher's Algebraic Topology P125. I will copy Hatcher's words and write down my questions in bold .$$$$ Let us find explicit cylcles representing generators of the infinite cyclic groups $H_n(D^n,\partial…
user12580
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How to determine image of the fundamental group of a covering space of $S^1 \vee S^1$

Consider the covering space of $S^1 \vee S^1$ in $(1)$. Then distinct loops in $(1)$ are represented by $\langle a, b^2, bab^{-1} \rangle$. Thus elements of the fundamental group are words generated by these distinct loops. This fundamental group…
user7090
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Connectivity and product

Let $X$ and $Y$ be two topological spaces. Is there any relation between Connectivity of $X\times Y$ and connectivity of $X$ and $Y$? I think by Kunneth formula, it must be $Conn(X\times Y)\leq\min\{Conn(X), Conn(Y)$}$ and I am wondering, How can be…
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Question about covering spaces

Suppose $\pi:M_1 \to M_2$ is a $C^\infty$ map of one connected differentiable manifold to another.And suppose for each $p\in M_1$,the differential $\pi_*:T_p M_1 \to T_{\pi(p)}M_2$ is a vector space isomorphism. (a)Show that if $M_1$ is…
Proton
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Fundamental group of the complement in $\mathbb R^3$ of a line and a circle

What is the fundamental group of the complement in $\mathbb R^3$ of a line and a circle. There are actually two cases to consider, one where the line goes through the interior of the circle and the other where it doesn?t. Are these two spaces…
user198206
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$X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. If $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$

Suppose $X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. I'd like to prove that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$. I've had the following thoughts: Let $f : X \to Y$ and $g: Y \to X$…
Matt
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How can a covering map (as defined in Hatcher) fail to be a surjection?

Hatcher defined a covering space as follows: $\textbf{Defn:}$ A covering space of a space $X$ is a space $\tilde{X}$ together with a map $p: \tilde{X} \to X$ satisfying the following conditions: There exists an open cover $\{ U_{\alpha}\}$ of $X$…
user7090
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Covering space and classification

Is there any classification for all topological spaces that have $\mathbb{R}^n$ as their universal cover?
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cohomology group of $SO(n)$

I am computing the Alexander-Spanier cohomology $H^i(SO(n),\mathbb{Z})$. I embedded $SO(n)$ into $R^{n^2}$. Since the embedding $i$ is a monomorphism, the induced group homomorphism $i^*$ is an epimorphism. Since $R^{n^2}$ is homotopic to a point,…
Qijun Tan
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Regarding an exercise of Hatcher (fundamental group of $\mathbb{R}^3$ with $n$ lines removed)

tl;dr I am not asking for a solution, that I already have. What do the generators for $\mathbb{R}^3 - \{n\text{ lines through the origin}\}$ look like explicitly, without making reference to any deformation retraction or homeomorphism? Exercise…
Andrew Thompson
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Trying to understand relative homology group

I'm reading about relative homology group but I'm having hard time in understanding this concept. So I was trying to find $H_1(D^n,S^{n-1})$, but I'm unable to solve this problem. Can someone give some idea for $n=1$ (Then I'll try to generalize…
Dontknowanything
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Given a triangulation (labeled complex), how do we determine the space?

Given a triangulation, how do we tell which space it is representing? My first idea would be to calculate the Euler Characteristic, but that would still leave some ambiguity, e.g. both the Projective Plane and the Disk have Euler Characteristic…
yoyostein
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