# 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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### How old is the distinction of right homotopy from left homotopy?

Going into the 1960s it seems to me that topologists saw path spaces as an advanced idea, useful in come contexts but not fundamental. So they took homotopy of maps as basically what is now called ...
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### turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
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### Theorem on space of maps to Eilenberg-Maclane space

I was reading the classic paper of Atiyah-Bott on Yang-Mills equations on Riemann surfaces. They mention a theorem attributed to Thom saying that if $X$ is a finite CW complex, then ...
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### Necessary condition for removing a simplex and changing homotopy type.

In a finite simplicial complex $K$, if the link of a simplex $\sigma$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type. I am wondering if ...
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### Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
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### Being contractible in homotopy theory vs. homotopy type theory

I'm trying to clarify the notion of being contractible in homotopy theory vs. homotopy type theory. Is the following right? "In homotopy theory the real interval $[0,1]$, considered as a subset ...
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### Bijection of homotopy classes

I want to prove the following: given the (already proven) fact that the we have a bijection between (continuous maps) $f:X\rightarrow Y^K$ and $g:X\wedge K\rightarrow Y$ for pointed spaces $X$,$Y$ and ...
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### Are $C^\infty$ exotic spheres $C^k$ exotic?

The only theory of exotic spheres that I know is of $C^\infty$ structures on them; that is, that there are plenty of spheres (in dimensions $n \geq 7$ that are homeomorphic but not diffeomorphic. To ...
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### Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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### Strong (trivial) cofibration in Lurie's HTT

in Lurie's book HTT in annexe A, proof of Proposition A.2.8.2 page 824, he mentions that a map is a "strong (trivial) cofibration" but I didn't succeed to find the definition of this notion that seems ...
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### Illustration of homotopies

While there are many pictures path homotopies, I fail to find any that illustrate normal homotopies (in the event "a normal homotopy" is something else, I clarify that I mean given two continuous ...
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### Resemblance between product and homotopy

The notion of product $X\times X$ for an object $X$ of a category $C$ resembles the notion of homotopy between two continuous functions. Indeed the relevant diagrams look the same: ...
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### Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$.

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$. $[X,Y]=\{f:X\to Y,f$ continuous $\}/\sim$ where $\sim$ is the homotopic equivalence. ...
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### Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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### Homotopy groups of infinite Grassmannians

Let $G_k$ be the infinite rank $k$ Grassmannian. For $n>k$, is $\pi_n(G_k)$ trivial? Phrased differently, is every rank $k$ vector bundle on an $n$-sphere, for $n>k$, trivial? (This is motivated ...
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### Proof that any 2 paths in a convex set are homotopic

A subset $A \in \mathbb{R}^n$ is convex if $\forall p,q \in A$, $\{{tp+(1-t)q | 0 \le t \le 1}\} \subset A$. I have two paths $\alpha, \beta : [0,1] \rightarrow A$, where $\alpha(0)=\beta(0)=p$ and ...
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### equivalence of n-connected spectra

Is the following claim true in general? Claim: A map $f:(X^r)\rightarrow (Y^r)$ of $n$-connective spectra (of simplicial sets) is a stable equivalence if and only if the corresponding map ...
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### Fibrations lift p-connectedness

Let $p: X \to B$ be a Serre fibration, and suppose that $B^p \subset B$ is a subspace of $B$ such that $(B,B^p)$ is $p$-connected, i.e. $\pi_n(B,B^p)=0 \ \forall n \leq p$ or, equivalently, ...
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### Show that g*c and c*g are homotopic, where g is a loop and c a constant loop

I've stumbled upon a question which asks me to prove that $f_0 := g*c$ and $f_1 := c*g$ are homotopic. More specifically it wants me to give a 'picture in $I\times I$, a picture in $X$ and an explicit ...
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### Calculating $\pi_2$ of a certain free loop space

For a topological space $X$, define $LX$ to be the set of continuous maps $S^1 \rightarrow X$ with the compact-open topology. Henceforth let $X = \Bbb{CP}^\infty \times \Bbb{RP}^\infty$, with ...
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### The space of $S^1/S^1$, the space of a single point, and their first homotopy group

I read from the book Soft matter physics by Kleman that the space $R$ of a point is $0$ and its first homotopy group $\pi_1(0)=0$. This causes some confusion to my understanding. Why the space of a ...
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### Why the space S1 and S1/Z_2 is topologically identical?

I am a physicist studying liquid crystals. My research is bit related to topology but I don't have much knowledge of it. Recently I read from a the book Soft matter physics: An introduction that ...
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### Hurewicz model structure and cofibrantly generated model categories

Is it an open problem if $\mathbf{TOP}$ with Hurewicz (Strøm) model structure is cofibrantly generated?
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### Complement of contractible subset of a sphere

Let $A$ be a nice closed subset of the sphere $S^n$; for example, we could ask $A\to S^n$ to be a cofibration. Assume that $A$ is contractible. Is then $S^n - A$ also contractible? It ...
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### Twisted cohomology as sections of bundle of Eilenberg-Maclane spaces

Let $X$ a space, and $E$ an multiplicative cohomology theory represented by a ring spectrum $K$, i.e. $E^\bullet(X)=[X,K]$. Also let $A$ be a local system of abelian groups. Cohomology with local ...
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### Checking the Homotopy of curves

Let $\gamma_0(t) = e^{2\pi it}$ and $\gamma_1(t) = e^{-2\pi it}$ for $0\le t\le 1$. I have to check whether or not they are homotopic in $\Bbb C -\{0\}$. I can see that the index of ...
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### A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
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### automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
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### Does the Seifert-van Kampen Theorem applied to loop spaces say anything about higher homotopy groups?

The Seifert-van Kampen tells us that $\pi_1$ preserves pushouts of the form $U \cap V \subseteq U, V$ (for $U \cap V$, $U$, $V$ open, path-connected). Can this information be used to say anything ...
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### About homotopy groups of pairs

Suppose $A$ is a deformation retract of $X$, for $n\ge 2$ and for any $x_0\in A$, how to show $$\pi_n(X,x_0)=\pi_n(A,x_0)\oplus\pi_n(X,A,x_0)$$ I am not clear why the homotopy groups are not ...
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### Maps that induces identity on fundamental groups are homotopic to identity?

Suppose $X$ is path connected, let $F:(X,x_0)\to (X,x_0)$ be a map such that $F_*: \Pi_1(X,x_0)\to \Pi_1(X,x_0)$ is identity, does it imply that $F$ is homotopic to identity? Let $y_0$ be arbitrary, ...
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### classifying space of permutation groups

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?
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### Mapping Cylinder.

I don't understand the following fact I've read: Any map $f:X \rightarrow Y$ can be written as composition $X \stackrel{i}{\hookrightarrow} M_f \stackrel{j}{\rightarrow} Y$, where $i$ is the ...
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### CW complexes - An algebraic Topology Question

This question regards a particular exercise regarding Algebraic Topology, CW complexes and homotopy. I am trying to prove that the Klein Bottle is homotopic to $S^2 \vee S^1 \vee S^1$, where $\vee$ ...
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### invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
### What is the map $\Sigma K(G,n) \to K(G,n+1)$?
Since $\Omega K(G, n+1)$ is a $K(G,n)$, we have a CW approximation/homotopy equivalence $K(G,n) \xrightarrow{\sim} \Omega K(G,n+1)$. The adjoint of this map is a map $\Sigma K(G,n) \to K(G,n+1)$. ...