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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How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups), and, as a system in my school, students must write one page about their theses explaining for non-mathematicians the purpose of the study (in general) and why it is important…
Rahman
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Why are homeomorphisms important?

I attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of algebraic topology is to classify surfaces in a way…
lithium123
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Why is a covering space of a torus $T$ homeomorphic either to $\mathbb{R}^2$, $S^1\times\mathbb{R}$ or $T$?

I know a sketch of the proof. M. A. Armstrong 's Basic Topology says that Suppose $X$ has a universal covering space, and denote it by $\tilde{X}$. Then the covering transformations form a group isomorphic to the fundamental group of $X$. Given any…
Roun
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Fundamental group of the special orthogonal group SO(n)

Question: What is the fundamental group of the special orthogonal group $SO(n)$, $n>2$? Clarification: The answer usually given is: $\mathbb{Z}_2$. But I would like to see a proof of that and an isomorphism $\pi_1(SO(n),E_n) \to \mathbb{Z}_2$ that…
Meneldur
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The homology groups of $T^2$ by Mayer-Vietoris

If I choose two open sets $A$ and $B$ as depicted on Wikipedia here: then I have an isomorphism between $H_n(A \cap B)$ and $H_n(A) \oplus H_n(B)$ because the two tubes in $A \cap B$ are disjoint. OK, so far so good. Then I write down the…
Rudy the Reindeer
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Geometry or topology behind the "impossible staircase"

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the so-called "impossible staircase"? Motivation: The…
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What is the topology of a world with portals?

Portal is a video game, where you can create 2 disks $D\in\mathbb{R}^3$, which then are identified. The world is glued together at these points. (source: thebuzzmedia.com) This kind of reminds me of some procedures to construct spaces for CW…
Nikolaj-K
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Prove that there is a two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are two Klein bottles in there, but how do I write down…
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An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$? We consider the real projective plane as a quotient of the disk. I didn't receive an exhaustive answer to this question from my teacher, in…
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Computing the homology and cohomology of connected sum

Suppose $M$ and $N$ are two connected oriented smooth manifolds of dimension $n$. Conventionally, people use $M\#N$ to denote the connecte sum of the two. (The connected sum is constructed from deleting an $n$-ball from each manifold and glueing the…
Honghao
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Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem

I have just learned about the Seifert-Van Kampen theorem and I find it hard to get my head around. The version of this theorem that I know is the following (given in Hatcher): If $X$ is the union of path - connected open sets $A_\alpha$ each…
user38268
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Do all continuous real-valued functions determine the topology?

Let $X$ be a topological space. If I know all the continuous functions from $X$ to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is somewhat artificial. So if this is wrong, will it be right if $X$ is a topological…
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CW complexes and manifolds

What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes? Thank you guys. This is just a question I thought of during class, obviously not homework...
LASV
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Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = \mathrm{colim}[\Sigma^kX,\Sigma^kY]$$ as a hom-set…
H. Shindoh
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What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many thanks in advance
El Moro
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