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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Have there been (successful) attempts to use something other than spheres for homotopy groups?

Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties: For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian. $\pi_n$ defines a functor from based spaces to (abelian)…
Najib Idrissi
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Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$

Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$ Show that there exist $[a,b]\subset [0,1]$, such that $$\int_{a}^{b}f(x) dx= \int_{a}^{b}g(x)dx = \frac{1}{2} $$ The question can be…
user622044
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Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable mathematical consequences of this theory so…
user02138
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What is knot theory about, exactly?

"In topology, knot theory is the study of mathematical knots." This is how Wikipedia defines knot theory. I have no idea what this is supposed to mean, but it does seem interesting. The rest of the article is full of examples of knots, their…
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Has anyone ever actually seen this Daniel Biss paper?

A student asked me about a paper by Daniel Biss (MIT Ph.D. and Illinois state senator) proving that "circles are really just bloated triangles." The only published source I could find was the young adult novel An Abundance of Katherines by John…
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$\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable

Question: Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable. Motivation: This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite get there. What I Can Do: There are proofs of this…
user2959
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How to interpret the Euler class?

Although I (hardly) understand the formal definition of the Euler class, I have very little intuition of it. I understand that the Euler class of $E\to X$ is zero if and only if there is a section, but what does it mean that the Euler class is…
Pierre D
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Research in algebraic topology

I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future: What else should I study after completing homology(basic), cohomology(basic) and homotopy…
K A Khan
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Meaning of relative homology

It is a bit easier to understand the homology $H_1(X, \mathbb Z)$ for various compact surfaces in analogy with handles and so on. There seems to be a nice intuitive picture with handles, holes, etc to think of the first homology group, and similar…
Quinn Rogan
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How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in passing that $X^n/X^{n-1}$ is the wedge sum of…
Dignaga
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Use of Reduced Homology

I've been reading Hatcher's Algebraic Topology, specifically the paragraph about reduced homology $\tilde{H}_*$ (for singular homology of topological spaces). Can someone please provide reasons why reduced homology is defined and studied? I…
Olivier Bégassat
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Homology of cube with a twist

Take the quotient space of the cube $I^3$ obtained by identifying each square face with opposite square via the right handed screw motion consisting of a translation by 1 unit perpendicular to the face, combined with a one-quarter twist of its face…
Juan S
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How are topological invariants obtained from TQFTs used in practice?

Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places: Atiyah, Topological quantum field theory Lurie, Topological Quantum Field Theory and the Cobordism Hypothesis Math Overflow,…
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"the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$, $S^{61}$"

This (long) paper, Guozhen Wang, Zhouli Xu. "On the uniqueness of the smooth structure of the 61-sphere." arXiv:1601.02184 [math.AT]. proves that the only odd dimensional spheres with a unique smooth structure are $S^1$, $S^3$, $S^5$,…
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Why is the Jordan Curve Theorem not "obvious"?

I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is obvious. Therefore answers like "do not trust…
Akash Kumar
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