# 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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### CW complex structure for $\mathbb{R}P^n$ or a space homotopy equivalent

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
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### Are there any nontrivial second Hurewicz homomorphisms for familiar compact 6-dimensional manifolds?

Based on several computations I have done, it seems that the second Hurewicz homomorphism $$h:\pi_{2}(X)\rightarrow H_{2}(X;\mathbf{Z})$$ has a habit of being trivial. For instance, this seems to be ...
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### multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $\circ$ on ...
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### Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
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### Are the space of paths with two given endpoints in a contractible space, contractible?

This question is inspired by an answer to Nitrogen's answer to my Are the path connected components of $\Omega S_1$ contractible? . Here we are asked whether the space of paths in $\mathbb{R}$ ...
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### Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has Tor-...
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### Are the path connected components of $\Omega S_1$ contractible?

Let $\Omega S_1$ be the space of loops of $S_1$ based at $x_0 \in S_1$ with the topology of uniform convergence. We know that the path connected components of this space are in one to one ...
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### Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
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### Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.

Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...
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### $T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
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### Gluing along infinitely many trivial cofibrations

I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps $f_\alpha: A_\alpha \to X$ each of which is a ...
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### Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
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### Show that $h$ is homotopic to the identity map relative to $C$.

This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3. 3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar ...
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### Why is the punctured plane not homotopic to the circle?

I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$. However, $X$ and $Y$ are not homotopic, i....
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### Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
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### How much algebra one needs to study algebraic topology and homotopy theory?

I wonder how much algebra(group theory, abstract algebra, linear algebra, ring theory, field theory) is assumed as a prerequisite in most of the modern algebraic topology text. For example, these J....
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### Trivial loop on the $1$-Skeleton

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is ...
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### projective model structure on presheaves , hom-functors are always cofibrant

Why hom-functors are always cofibrant in the projective model structure in $[\cal T,\cal V]$? The claim is here on page 5.
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### Fixed point property of spaces having same homotopy type

Suppose X and Y have same homotopy type.X as a topological space has fixed point property.Can we conclude anything about fixed point property of Y?
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### How to show a straight line homotopy is continuous?

Given $f$ and $g$ continuous maps from $X$ into $\mathbb{R}^{2}$, how to show that the straight line homotopy map $F(x,t)=(1-t)f(x)+tg(x)$ is continuous?
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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 ...
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 ...