# Tagged Questions

Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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### Universal covering group and fundamental group of $SO(n)$

The universal cover of $SO(2)$ is $\mathbb{R}$, whilst the fundamental group is $\mathbb{Z}$. That is $$SO(2) \cong \mathrm{universal\ cover}/\pi_1$$ Likewise, I believe that the universal cover of ...
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### Why does sheaf $\pi_0$ of a simplicial presheaf determine maps to sheaves?

This question refers to an argument from Section 6, p. 22 of Freed–Hopkins, "Chern–Weil forms and abstract homotopy theory." There's something presumably straightforward I'm just not ...
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### Good reference for “solving equation $f(x)=0$ by homotopy and continuation methods”

I need a good reference for "solving equation $f(x)=0$ by homotopy and continuation methods". If $f:X\to Y$ is a continuous map between to linear space $X$ and $Y$, we want to find the roots of $f$. ...
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### How to prove that the set of real $n \times n$ PSD matrices of rank $\leq r < n$ and unit trace, is not contractible?

Extended Version of the Question: How to prove that the set of real $n \times n$, symmetric positive semidefinite (PSD) matrices of rank $\leq r$ ($1 \lt r \lt n-1$) and unit trace, is not ...
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### Eilenberg-MacLane Spaces $K(G,n)$: Construction!

I'm looking for an easy construction of $K(G,n)$, Eilenberg-MacLane spaces. I know I can use the Postnikov Towers for the upper part $\pi_i(X)=0$ for $i > n$. For the lower part $\pi_i(X)=0$ for ...
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### Fundamental group of a circle with rational lines

Let $X$ be the subset of $\mathbb{R}^2$ given by the union of the unit circle the $y$-axis all lines through the origin with rational slopes equipped with the subspace topology. Is there a simple ...
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### Is this 2-complex a $K(\pi,1)$?

Consider $CW$-complex $X$ obtained from $S^1\vee S^1$ by glueing two $2$-cells by the loops $a^5b^{-3}$ and $b^3(ab)^{-2}$. As we can see in Hatcher (p. 142), abelianisation of $\pi_1(X)$ is trivial, ...
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### Why is symmetric group action needed for symmetric spectra?

I know that Boardman spectra aren't supposed to have an on-the-nose commutative smash product, and symmetric spectra -- which look like essentially the same thing, except the spaces $X_n$ have to come ...
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### Long Exact sequence of Relative Homotopy Groups: examples and applications

I'm going to make a talk around higher homotopy groups, and the long exact sequence of relative homotopy groups. I would like to show some nice examples and applications of this theorem after the ...
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### Parallely transported vector continuous with respect to homotopy of curve?

Let $M$ be a smooth manifold with connection. Choose two points $p,q \in M$ and connect them with curve $\gamma _0$. Suppose I take fixed vector at $x$ and parallel transport it to $y$. Denote this ...
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### Is the smash product of two Moore spaces again a Moore space?

Write $M(G,n)$ for the Moore space with $\tilde{H}_\ast(M(G,n);\mathbb{Z})$ naturally isomorphic to $G$ concentrated in degree $n$. Now fix finitely generated (Abelian) groups $G$ and $H$ and ...
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### Proof of $BGl(n)\simeq Gr(n,\infty)$

What is a direct proof for the existence of a weak homotopy equivalence between the Grassmanian $Gr(n):=Gr(n,\infty)$ and the classifying space $BGL(n)$ of $GL(n)$? They both represent the ...
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### Van Kampen theorem application in a simple three-holed figure

The purpose of this question is to understand the computations to get the expression of the fundamental group in a simple case using the Van Kampen theorem. Let $X$ be the three holes object ...
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### Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
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### how to evaluate homotopy group of this specific structure

I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure. In a paper, a specific topological structure is defined. The structure ...
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### Using sequence of least squares solutions to solve non-linear problems?

I do know about the iteratively reweighted least-squares and have played around with it to some success finding non-linear solutions (like minimizing non-2-norms to achieve solutions which seem to be ...
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### The fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed

Determine whether the fundamental group of $\mathbb{R}^3$ with its non-negative half-axes removed is trivial, infinite cyclic, or isomorphic to the figure eight space. I found this answer: ...
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### Theta-space is a deformation retraction of the doubly-punctured plane, how to find equations.

That theta space is given by $S^1\cup(0\times[-1,1]) \subset\mathbb{R}^2$ it is said that this space is a deformation retract of the doubly punctured plane, here is the explanation I found: The ...
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### The fundamental group of $B^2\times S^1$

In one exercise we are supposed to find the fundamental group of $B^2\times S^1$. It is given that the fundamental group is $\mathbb{Z}$, because we can show that $S^1$ is a deformation retract of ...
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### Homotopy cardinality of the category of categories

The category of finite sets has homotopy cardinality $e$, because $$|{\bf FinSet}|=\sum_{n=0}^{\infty}\frac{1}{\left|\operatorname{Aut}\ [n]\right|}=\sum_{n=0}^{\infty}\frac{1}{n!}.$$ What is the ...
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### $h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic.

I want to show that if $h,h':X\to Y$ are homotopic and $k,k':Y\to Z$ are homotopic, then $k\circ h$ and $k'\circ h'$ are homotopic. This means that there is a continuous map $F_1:X\times I \to Y$ ...
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### Does a weak homotopy equivalence induce an equivalence of categories on the fundamental groupoids?

Let $f\colon X\rightarrow Y$ be a weak homotopy equivalence. ($\pi_0(f)$ is a bijection and $\pi_n(f,x)$ is an isomorphism for all basepoints $x\in X$ and all $n$.) It induces a functor ...
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### Where is basic algebraic topology in basic algebraic geometry?

I'm a student meeting commutative algebra and algebraic geometry for the first time. The idea of studying every (commutative) ring geometrically via its spectrum (as a locally ringed space) is ...
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### Homotopy commutative diagrams and homotopy equivalent spaces

The question is fairly general. Suppose I have a homotopy commutative diagram of the form \require{AMScd} \begin{CD} A @>{f}>> B\\ @V{h}VV @V{i}VV \\ C @>{g}>> D ...
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### Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
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### One point union, second homotopy group is not finitely generated?

Let $X$ be the one-point (wedge sum) union of the circle $S^1$ and the sphere $S^2$. What is the easiest way to see that the abelian group $\pi_2(X)$ is not finitely generated?
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### Free Product of Groups with Presentations

There is a highly believable theorem: Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations ...
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### Homotopy fixed points of connective K-theory

Let $ku$ be the $p$-completion of the connective complex K-theory spectrum. The group $\Bbb{Z}_p^\times\cong \Delta \times \Bbb{Z}_p$ acts on $ku$, where $\Delta$ is the cyclic group of order $p-1$. ...
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### Conjugate paths have free homotopic circle representations?

Is this statement true? In a path connected space $X$, conjugate elements of $\pi_1(X,p)$ have free homotopic circle representations. This is related to my other question here. Basically, I am ...
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### Where to learn about model categories?

A model category (introduced by Quillen in the sixties) is a category equipped with three distinguished classes of morphisms (weak equivalences, fibrations and cofibrations) satisfying some axioms ...
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### When should one learn about $(\infty,1)$-categories?

I've been doing a lot of reading on homotopy theory. I'm very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the ...
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### Bijection induced by mapping loops to their circle representations

Let $X$ be a path connected space, and $p \in X$. I wish to show that the map $f \mapsto \tilde{f}$ induces a bijection between the conjugacy classes of $\pi(X,p)$ and $[\mathbb{S}^1: X]$, the free ...
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### $M$ is homotopy equivalent to $S^n$. [duplicate]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. How do I see that $M$ is ...
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### Does it follow that $Y$ is homotopy equivalent to $S^{n-1}$?

Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does it follow that $Y$ is ...
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### Homotopy fixed points for product of groups

I want to show the following: Let $G$ and $H$ be groups, and let $X$ be a $G\times H$-space/spectrum. Then, $(X^{hH})^{hG}\simeq X^{h(G\times H)}$ with the obvious actions of $G$ and $H$ on $X$. I ...
I'm trying to calculate the fundamental group of the following space: I've been thinking that I should apply Seifert - Van Kampen theorem but I haven't been able to choose some nice open sets $U$ ...
### Compact $n$-manifold has same integral cohomology as $S^n$?
Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does $M$ have the same integral ...