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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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Universal cover of wedge product of circles

I want to ask a question about universal covering of wedge space of two circles. It is known that the universal covering space is the cayley graph. I have another thing in mind which I came up with before seeing the Cayley graph as the answer to…
Sina
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Two different kinds of actions on fibers of a universal cover

This concerns Hatcher's exercise 1.3.26. It says almost this: Given a universal cover there are two actions of $\pi_1$ on the fibers, one given by lifting loops and one given by restricting deck transformation to fibers. Are these equivalent ? When…
sifsa
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Interpretation of boundary homomorphism in long exact sequence of homology groups

In Allen Hatcher's book on page 117 (bottom) he says the following: The boundary map $\partial : H_n (X, A)\rightarrow H_{n−1} (A)$ has a very simple description: If a class $[\alpha] \in H_n(X,A)$ is represented by a relative cycle $\alpha$,…
val11
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Why is $\mathbb R^n$ so important in algebraic topology?

If you study the different tools from algebraic topology, you realise that most (if not all) of them are somehow meant to compare the space with Euclidean space: Homotopy theory deals with maps from $S^n\subset\mathbb R^{n+1}$ into the…
Gaussler
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Homology Group of Quotient Space

Let X be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. Do the same for $S^3$ with antipodal points of the equator $S^2 \subset S^3$ identified. This is probably…
shay
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Computing this fundamental group

What is the fundamental group of $$X = \left\{\left(\sqrt{x^2+y^2}-2\right)^2 + z^2 = 1\right\}\cup \left\{(x,y,0)\;\; :\;\; x^2 + y^2 \leq 9\right\}\subset\mathbb R^3\,?$$ I would say that it is $\,\mathbb Z\,$ cause you can deform one of "the…
thetruth
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The fundamental group of $\mathbb S^2$ attached with a diameter?

What is the fundamental group of $\mathbb S^2$ attached with a diameter? And what is the fundamental group of a hemisphere attached with a diameter? I guess for the latter one, we can deformation retract it to a circle by compressing around the…
user198206
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Fundamental group of $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$

Calculate the fundamental group of the complement in $\mathbb{R}^3$ of $$\{ (x,y,z) \ | \ y = 0 , \ x^{2} + z^{2} = 1\} \cup \{ (x,y,z) \ | \ z = 0 , \ (x-1)^{2} + y^{2} = 1\}.$$ Note: this space is $\mathbb{R}^{3}\setminus \{ \mbox{2 linked…
WLOG
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topology of $PSL(2, \mathbb C)$?

What can one say about the topology of $\text{PSL}(2, \mathbb{C})$, for example its cohomology groups? By using the long exact sequence in homotopy one can show that the inclusion $$\text{SO}_3 \to \text{PSL}(2, \mathbb{C})$$ that sends any rotation…
Casaubon
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what is the degree of $f :S^n \to S^n$ when $f$ has no fixed points?

Let $n \ge 1.$ and let $f: S^n \to S^n$ be continuous self-map of the unit $n$-sphere. If $f$ has no fixed points, what is the degree of $f$ , and why? Thanks in advance.
Danny
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Ring structure of $H^2(S^2 \vee S^4)$

We know that $H^p(S^2 \vee S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \vee S^4)$ I want to cup it with itself and get 0. My idea is to use…
M.B.
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All regular covering projections of wedge product of $\mathbb{RP}^3$ with $S^1$

I'm studying for a topology exam. One question in an exam paper from a previous year is: Starting with the universal cover, describe all the regular covering projections of $\mathbb{RP^3}\vee S^1$. The universal covers of $\mathbb{RP^3}$ and $S^1$…
Stu
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$S^1 \times S^2$ vs $S^1 \vee S^2 \vee S^3$

This is a multi-part problem. Let $X = S^1 \times S^2$ and $Y = S^1 ­\vee S^2 \vee S^3.$ Compute $\pi_1$ of those spaces. Do there exist $\phi:S^3 \to X$ and $\psi:X \to S^3$ such that $\psi \phi \simeq 1_{S^3}?$ Hint: use covering space theory and…
anon
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Homotopy lifting property of $\mathbb{R} \to S^1$ in Hatcher

I am reading Hatcher's proof of the homotopy lifting property of the covering map $p: \mathbb{R}\to S^1$. Starting with a homotopy $F: Y \times I \to S^1$ and a map $\tilde{F}:Y \times \{0\} \to \mathbb{R}$, first he shows existence of the lift for…
Eric Auld
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Does homotopy equivalence of pairs $f:(X,A)\to(Y,B)$ induce the homotopy equivalence of pairs $f:(X,\bar A)\to(Y,\bar B)$?

When we have a homotopy equivalence through a pair $f:(X,A)\to (Y, B) $, it is said that we can induce a homotopy equivalence through a pair $f:(X,\bar A)\to (Y,\bar B) $, where $\bar A$ stands for the closure of A. Do you know how we can prove…
Emily
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