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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3
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24 views

Postnikov towers for non-CW spaces

In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary. Question: Does ...
0
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0answers
14 views

Proof of the HELP theorem (Homotopy Extension and Lifting property)

I'd like to read a proof of the HELP (Homotopy Extension and Lifting Property) Thereom with as many details as possible ! Is there any books/documents that has it ? Thanks
-1
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0answers
22 views

cohomology homomorphism induced by classifying map [on hold]

(1). Prove that there exists a principal $Sp(1)(\cong S^3)$-bundle over $\mathbb{C}P^\infty$, denoted as $Sp(1)\to E\to \mathbb{C}P^\infty$, such that $E\simeq S^2$. (2). The universal bundle ...
1
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1answer
25 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
0
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0answers
18 views

fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
0
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0answers
12 views

Proof that $ X \times I$ is a CW complex

I am trying to prove that the mapping cylinder is a CW complex and to start, I need to show that $ X \times I$ is a CW complex, where $X$ is a CW complex . I haven't seem the proof that the product of ...
0
votes
1answer
25 views

Show that the cylinder is not ambient isotopic to the Mobius band.

Here is my definition for ambient isotopy: We say if there is an orientation preserving piecewise linear homeomorphism $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ (or replace $\mathbb{R}^3$ with $S^3$) ...
-1
votes
1answer
78 views

Homotopy Groups for Categories

With this observation in mind how far are we from defining $\forall \mathcal{C} \ \text{category}\ \pi_1(\mathcal{C})$? Let me be more clear. Let be $n$ the following category $0 \rightarrow 1 ...
2
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0answers
55 views
+250

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
1
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0answers
34 views

Homotopy equivalence of pushouts of topological spaces

Let $h \colon A \to B$ and $r \colon S^{n-1} \to A$ be continuous maps. Assume that $h$ is an homotopy equivalence, prove that $$ D^n \cup_{r} A \simeq D^n \cup_{h \circ r} B$$ where $D^n ...
0
votes
0answers
26 views

Are $z^n$ and $p(z)/|p(z)|$ homotopic?

Let $p:\mathbb C\longrightarrow \mathbb C$ be a complex polynomial with no zeros and degree $n$. Is it true that the maps $f, g:S^1\longrightarrow S^1$ given by $$f(z)=z^n\quad \textrm{and}\quad ...
1
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1answer
21 views

Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where ...
2
votes
1answer
26 views

Fundamental group of sphere

To show that $S^3$ is not diffeomorphic to $S^2 \times S^1$, I'd like to say that their fundamental groups are not the same. So $\pi_1(S^3)= 0 $ but why is $\pi_1(S^2 \times S^1) = Z $ ?
2
votes
1answer
63 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...
3
votes
1answer
23 views

How to show that homotopy of chain maps respects composition?

Given the homotopic pairs of chain maps $f_1 \simeq f_2 : A_* \to B_*$ and $g_1 \simeq g_2 : B_* \to C_*$, show that $g_1 \circ f_1 \simeq g_2 \circ f_2: A_* \to C_*$. $f_1 \simeq f_2$ means that ...
1
vote
1answer
60 views

On chain homotopy equivalence

I just learnt the notion of chain map and have the following question. Let $C=(C_n,\partial_n^C)$ and $D=(D_n,\partial_n^D)$ be chain complexes of abelian groups with boundary maps $\partial_n^C$ and ...
2
votes
1answer
132 views

Question about homotopy equivalence

I have this proof but I don't understand why $i\circ j$ induces a homotopy equivalence, and how to see $j_*$ is injective at the level of homology? $X$ is a Banach space
0
votes
1answer
34 views

Proving that induced homomorphism is an isomorphism

Let $A \subset X$ and let $j: A \to X$ be the inclusion map. Let $f: X \to A$ be a continuous map. We suppose there is a homotopy between $j \circ f$ and the identity map on $X$. We are to show that ...
4
votes
0answers
33 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
1
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0answers
28 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
votes
1answer
31 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
votes
1answer
17 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
votes
1answer
32 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
votes
0answers
30 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
58 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 ...
1
vote
1answer
61 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
votes
2answers
54 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
votes
1answer
46 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 ...
0
votes
0answers
11 views

stable splittings of projective space

On Hatcher's book Algebraic Topology, page 468 Prop. 4I.3, For prime number $p$, can we decompose $\mathbb{C}P^\infty$ in a similar way?
0
votes
0answers
17 views

Solving large non-linear polynomial equation system

I have a 2 order equation system of 7 unknowns. It is constructed as this: F1=0,F2=0,F3=0...F7=0 of which F1=f1*f2,F2=f3*f4... And f1=a1*p1+a2*p2+a3*p3+a4*p4+a5*p5+a6*p6+a7*p7 a1~a7 are known ...
4
votes
2answers
55 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
votes
1answer
28 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
votes
3answers
40 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 ...
1
vote
1answer
40 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
votes
1answer
32 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} ...
0
votes
0answers
18 views

Making a homotopy equivalence out of a pushout map involving $A \cup B = X$

Given the topological space $X$ with subspaces $A$, $B$ so that $A \cup B = X$ and the maps in the "square" of the following diagram ($i_1$, $f$, $g$, $h$) forming the pushout $Y$: I added rest of ...
1
vote
0answers
26 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
73 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 ...
0
votes
1answer
46 views

maps between spheres, torus and projective plane [closed]

How to solve these questions by direct and valid argument? Various methods are wanted. Thanks.
1
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1answer
59 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 ...
0
votes
0answers
30 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
1
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1answer
29 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 ...
5
votes
1answer
43 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 ...
1
vote
0answers
37 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
votes
1answer
41 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 ...
1
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1answer
41 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
votes
1answer
79 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
votes
1answer
29 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
votes
1answer
26 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 ...
2
votes
1answer
36 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 ...