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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72 views

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 ...
0
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1answer
94 views

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 ...
0
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1answer
51 views

Homotopy groups of pairs and homotopy fibration of inclusions

Let $(X,A)$ be a pair of topological spaces, where $X$ is path-connected and $A$ is a path-connected subespace of $X$ with a base point. So, we have a long exact sequence of homotopy groups $$... \to ...
0
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1answer
41 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
0
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0answers
56 views

Invertibility of suspension in spectra

I know that spectra are supposed to be designed so that suspension is invertible up to homotopy, but I'm having trouble articulating exactly why this is the case. If $E$ is a spectrum and $\Sigma E$ ...
3
votes
2answers
144 views

Homology of $n$-sheeted covering space

Let $X$ be the Klein bottle, that is $X=\mathbb{R}^2/G$ with $$G=\langle a,b\mid a^{-1}b ab=1\rangle,$$ acting via $a: \mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto (x+1,y)$, $b: \mathbb{R}^2\to ...
0
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1answer
263 views

cohomology of suspension

Let $X$ be a topological space. Let $\Sigma$ be suspension. Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not? Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma ...
3
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2answers
77 views

Does Homotopy Equivalence Lead to a Homeomorphism?

I've read online that "Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another (i.e., made homeomorphic) by bending, shrinking and expanding operations", ...
3
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1answer
168 views

homology of smash product of Eilenberg-Maclane spaces

Let $K_n=K(\mathbb{Z},n)$ be the Eilenberg-Maclane space. Prove: (1). $K_m\wedge K_n$ is $(m+n-1)$-connected. (2). $H_{m+n}(K_m\wedge k_n;\mathbb{Z})= H_{m+n}(K_m\times k_n;\mathbb{Z})$. How to ...
4
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2answers
95 views

Is $H^*(\mathbf{C} P^\infty)=R[X]$ or $R[[X]]$?

The first ring seems to be what one learns first: the underlying group is the cohomology of the total singular cochain complex $C^*(\mathbf{C} P^\infty)$, which is defined as $\oplus C^n(\mathbf{C} ...
2
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1answer
63 views

$E_{\infty}$ spaces are $A_{\infty}$ spaces

While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...
0
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3answers
168 views

Homotopy equivalence for $S^n$ with finite k punctures

I need help determining to what $S^n/${k points}--the n-dimensional sphere missing a finite k number of points-- is homotopy equivalent. I tried envisioning the above for n=2: $S^2/${1 point} is ...
2
votes
1answer
230 views

Homotopy equivalences in pushout square with cofibration.

If following square is a pushout square, $g$ is a cofibration and $f$ is a homotopy equivalence then $i$ is also a homotopy equivalence. $$ \begin{matrix} A & \xrightarrow{f} & B \\ ...
3
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1answer
53 views

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 \begin{equation} ...
1
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0answers
40 views

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 ...
2
votes
1answer
89 views

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 ...
2
votes
1answer
176 views

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 ...
1
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1answer
54 views

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 ...
6
votes
1answer
81 views

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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0answers
173 views

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: ...
0
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1answer
63 views

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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1answer
55 views

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 ...
3
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1answer
110 views

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: ...
0
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1answer
36 views

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. ...
0
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1answer
44 views

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 ...
5
votes
1answer
219 views

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 ...
0
votes
1answer
57 views

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 ...
1
vote
0answers
24 views

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 ...
1
vote
1answer
100 views

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, ...
1
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1answer
39 views

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 ...
6
votes
1answer
103 views

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 ...
1
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1answer
31 views

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 ...
3
votes
2answers
112 views

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 ...
3
votes
1answer
120 views

Hurewicz model structure and cofibrantly generated model categories

Is it an open problem if $\mathbf{TOP}$ with Hurewicz (Strøm) model structure is cofibrantly generated?
2
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0answers
57 views

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 ...
0
votes
0answers
60 views

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 ...
0
votes
1answer
34 views

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 ...
3
votes
1answer
159 views

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), ...
6
votes
0answers
118 views

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 ...
1
vote
3answers
167 views

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 ...
1
vote
2answers
40 views

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 ...
2
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2answers
84 views

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, ...
1
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1answer
72 views

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 $\Sigma_n$?
3
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0answers
39 views

How can I prove that if for all $X \in \Bbb{Top_*}$ , $[X,W]_*$ has a natural group structure, then $W$ is an $H$-group?

I know that if $\enspace[X,W]_*$ has a natural group structure, in particular $\enspace[W \times W, W]_*$ has it. If $p_1,p_2:W \times W \to W$ are the projections, it seems that defining $ \mu : W ...
5
votes
2answers
105 views

Maps to Sn homotopic

At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) ...
0
votes
1answer
257 views

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 ...
1
vote
1answer
73 views

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 $ ...
1
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1answer
206 views

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 ...
1
vote
1answer
215 views

Every Group is a Fundamental Group

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 ...
0
votes
2answers
90 views

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 $\Sigma K(G,n) \to K(G,n+1)$. ...