# Tagged Questions

The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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### Proof of the bigon criterion

In Farb and Margalit's A Primer on Mapping Class Groups we have the following Proposition 1.7: Two transverse simple closed curves in a surface $S$ are in minimal position iff they do not form a ...
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### universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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### Connected sum of two “same” Klein bottles

If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles? This is what ...
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### What is the distance of two circles put on eachother?

If one takes two circles (lets say on a straight cilinder), and bring the circles closer and closer to eachother. Will the distance between them goes to zero, or can you say the distance is for ...
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### Unbounded, closed, star-shaped set contains ray

I am trying to prove the following statement: Let $R$ be a real closed field (such as the real numbers). Let $M\subseteq R^n$ be a semi-algebraic set, i.e. a set which is defined by a Boolean ...
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### How to prove, no tame knot is isotopic to a non tame knot?

Please let me ask the following question. I have read in Wikipedia, quote: A polygonal knot is a knot whose image in R^3 is the union of a finite set of line segments. A tame knot is any knot ...
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### Topological Equivalence of Metric Spaces [closed]

Suppose we have two different metric spaces $(X,\phi)$ and $(Y,\psi)$. I need to show that the metrics $\phi$ and $\psi$ are equivalent metrics. Using a sterographic projection, I've shown that if we ...
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### A. A. Markov's paper on insolubility of the homeorphy problem [duplicate]

I am aware that this has been asked before, but the paper is nowhere to be found online, the provided link in the old thread leads to nowhere, and I'm really at wits end to find this paper, can anyone ...
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### Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...