# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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### Knot Group and the Unknot

Hi I am stuck in trying to show that given a knot $K$ such that the knot group $\pi_1(K)=\mathbb Z$ then $K\simeq U$. I tried to use the fact that the infinite cyclic cover is the universal cover but ...
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### What are polyhedrons?

Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ ...
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### Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
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### Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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### how to show whether this topological space is metrizable or not?

Let $X$ be a two-element topological space with a discrete topology. Let $J$ be an uncountable indexed set. And let $Z=X^J$ be the Cartesian product endowed with the product topology. Is $Z$ ...
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### $\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$ My question : ...
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### Weak hausdorff but not compactly generated? [duplicate]

What is an example of a weak Hausdorff space that is not compactly generated? I can't think of any, and googling doesn't reveal anything...
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### Self study Persistent Homology

I am a graduate student in mathematics interested in Persistent Homology. Can anyone recommend any good books / resources to self study Persistent Homology? I am taking a course in Algebraic ...
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### Proof of : “Signature of $\mathbb{C}P^{2n}$ is $1$”

I started learning about signature of a $4k$-manifold and one of the most common example is the signature of $\mathbb{C}P^{2n}$. The only reference I found is tom Dieck's Algebraic Topology. Even ...
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### Connect Sum of a connected, compact manifold of dimension n and $S^n$

$M$ be a connected,compact manifold of dimension n. Show that $M \# S^n$ is homeomorphic to $M$ My idea: $S^n-D^n$ is homeomorphic to $D^n$..so $M \# S^n$ is homeomorphic to $(M-D^n) \cup D^n$ ......
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### Triangle inequality of hyperbolic metric

For $z_1, z_2 \in \mathbb{B}^2$, define $d(z_1, z_2) = \text{cosh}^{-1}(1+ \dfrac{2|z_1 - z_2|^2}{(1-|z_1|^2)(1-|z_2|^2)})$. In my text book (Lee's Topological manifolds Problems 12-23), to prove ...
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### Are all paths with the same endpoints homotopic in a simply connected region?

It is clear to me that if all paths (with the same endpoints) in a region are homotopic then that region is simply connected, however I am having difficultly proving the converse, that is, all paths ...
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### Simpicial approximation map

Let $\Delta^n$ be the standard $n$-simplex and $f :\delta\Delta^n\to \delta\Delta^n$ ($\delta\Delta^n$ is a boundary of $\Delta^n$ ) be a continuous function such that $f(-x) = -f(x)$. Is there a ...
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### Definition of General Position for a Semi-Algebraic Set

I'm looking for a precise definition of what it means for a semi-algebraic set to be in general position. I've found definitions that apply to sets of points, but that doesn't seem to help in the ...
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### How can I draw (using a computer) spaces that I can't parametrize easily?

I am studying algebraic topology and I came around the following problem: I have to describe the space obtained when I identify the circles marked with different letters in the following figure: ...
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### Euler number zero for odd dimensional compact manifolds

I need to prove that every compact manifold of odd dimension has Euler number zero. The Euler number of $M$ compact and oriented is $$e(M):=\int_Ms_0^*\phi(TM)$$ where $s_0$ is the zero section of ...
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### Two topological groups $\mathrm{O}(n)$ (orthogonal group) and $\mathrm{SO}(n)\times \mathbb{Z}_2$

Problem. (Basic topology (M.A.Armstrong) Exercise 16 in chapter 4.3) (1) Prove that $\mathrm{O}(n)$ is homeomorphic to $\mathrm{SO}(n)\times \mathbb{Z}_2$. (2) Are these two isomorphic as topological ...
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### Problem understanding how to compute fundamental group of connected sum of torus

I have attempted trying to compute the fundamental group of a 2 torus, however I don't know how to proceed to "simplify" the result after applying van Kampen's Theorem. I calculated the fundamental ...
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### Fundamental Group of the Space X

May I know what is the name of this space $X$ obtained from the picture above? Also, what is its fundamental group? I tried calculating by triangulation and using a algorithm involving maximal trees, ...
If $\Gamma$ is a proper finite index subgroup of $PSL_2(\mathbb{Z}) \cong C_2*C_3$, then must there exist a $\Gamma'$ finite index in $\Gamma$ such that $\mathcal{H}/\Gamma'\rightarrow\mathcal{H}/\... 1answer 78 views ### How to prove that$T^1(M)$is simply connected for some specific$M$. Concretely, I'm working with the spaces:$S^n$,$\mathbb{C}P^n$and$\mathbb{H}P^n$. I need to conclude that$T^1(M)$is simply connected for all those manifolds$M$I listed (with the exception of$...
Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that ...