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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3answers
50 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
0
votes
0answers
19 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
0
votes
0answers
14 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
1
vote
0answers
11 views

co H-space on CW-complex [on hold]

Does someone have any idea how to prove these? If X is (n-1)-connected CW-complex of dimension <2n then there exists homotopy co-multiplication on X. If X is (n-1)-connected CW-complex and ...
2
votes
1answer
15 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
0
votes
1answer
34 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
0
votes
1answer
54 views

Homotopy equivalence?

Can someone explaine what this means mathematicaly : "Let us denote by $h: X\rightarrow Y$ a homotopic equivalence map for which $h|_{Y}$ is the identity " Remark: $Y$ is include in $X$ Please ...
1
vote
0answers
15 views

use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
-2
votes
1answer
42 views

what does Homotopy Tell?

What is the Homotopy geometrically?And what is path-homotopy? If two real valued functions are homotopic to each other, then what can we say about the geometry behind this? It need not be equal ...
1
vote
0answers
13 views

$S^1$ a p-local complex?

Let $p$ be a prime. Is $S^1$ a p-local CW-complex? Meaning, for any reduced homology theory $\overline{E}_*$, do we have $\overline{E}_*(S^1)=\overline{E}_*(S^1) \otimes_{\mathbb{Z}} ...
2
votes
0answers
45 views

Reduced homology and colimits

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then $$ \displaystyle\tilde H_k(X) = ...
0
votes
0answers
23 views

Serre fibration and CW-complexes

Suppose that $p:X\rightarrow E$ is a Serre fibration. I know the definition: then $p$ has the right lifting property with respect to all inclusions $I^n\rightarrow I^{n}\times I$. Now it seems that ...
0
votes
1answer
59 views

Which of these are homotopy equivalent? $S^1, \mathbb{R}, \{*\}$

Which of these spaces are homotopy equivalent: $S^1, \mathbb{R}, \{*\}$? I found a homotopy equivalence between $\mathbb{R}$ and the one point space $\{*\}$, so they are homotopy equivalent. The ...
1
vote
2answers
44 views

Proving that a continuous map is homotopic to the constant map

How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be ...
0
votes
0answers
22 views

Is $EG \times EG \times Y$ homotopy equivalent to $EG \times Y$ through $G$-maps?

This is loosely related to my previous question Homotopy type vs. weak homotopy type, and repercussions for EG, and more closely related to the answer to the question Contractibility vs. ...
1
vote
1answer
22 views

Lifting properties of Serre fibrations

Suppose that $p:X\rightarrow B$ is a Serre fibration. I want to prove that $p$ has the right lifting property with respect to all maps of the form: $$S^{n-1}\times ...
1
vote
0answers
25 views

conjugation of Lie groups and homotopy group

Let $G$ be a Lie group. Let $\phi, \psi \in \pi_n(G)$. Consider $\theta \in \pi_n(G)$ defined by $\theta(x):= \phi(x)\psi(x)\phi(x)^{-1} \in G$, where we use multiplication and inversion in $G$ in ...
0
votes
1answer
33 views

Extending a Homotopy from $X \times [0,1]$ to $X \times \mathbb{R}$

In Jeffery M. Lee's Manifolds and Differential Geometry Exercise 1.77: For smooth manifolds $X$ and $Y$, show that if $f_0: X \rightarrow Y$ and $f_1: X \rightarrow Y$ are $C^r$ homotopic then ...
3
votes
0answers
79 views

The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
1
vote
1answer
49 views

Adjoining an $(n+1)$-cell is an $n$-equivalence

Suppose $X$ is a topological space and $x_0 \in X$. Let $$ X' = X \cup e^{n+1} $$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the ...
0
votes
0answers
90 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
0
votes
0answers
21 views

Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...
4
votes
2answers
74 views

Homotopy of maps $(D^n, S^{n-1}) \longrightarrow (X,A)$ relative $S^{n-1}$

Suppose that for a map $f: (D^n, S^{n-1}) \to (X,A)$ (where $(X,A)$ is an arbitrary pair of spaces) there exists a homotopy $H: D^n \times I \to X$ with $H(\_,0)=f$, $H(s,t) \in A$ for $s \in S^{n-1}$ ...
1
vote
0answers
30 views

Retraction and intersection

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically ...
2
votes
0answers
26 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get ...
1
vote
0answers
32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
1
vote
1answer
42 views

Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
2
votes
0answers
23 views

Does the 2-functor $PsAlg\to \mathfrak{X} $ reflect equivalences?

Consider a $2$-monad $ T: \mathfrak{X}\to \mathfrak{X} $ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $ U: PsAlg\to \mathfrak{X} $. Does this forgetful functor ...
1
vote
0answers
53 views

Hopf invariant and homotopy groups of spheres

I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$. I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n}) ...
2
votes
0answers
35 views

Sequence of cofibrations gives strict commutativity of given triangle

The problem is the following: Suppose $X_0\rightarrow X_1\rightarrow\cdots$ is a sequence of cofibrations. We will denote $f_n:X_n\rightarrow X_{n+1}$. Assume that we have also maps ...
1
vote
0answers
37 views

Null-homotopic maps

Assume that $[\alpha]\in\pi_n(X,x_0)$. I want to prove the following: $[\alpha]=0$ if and only if $\alpha:S^n\rightarrow X$ extends to a map $D^n\rightarrow X$. Can someone help me with this proof? ...
0
votes
1answer
48 views

Homotopy Type of Surface of Genus g

Need help with the following exercise; "Let M be a compact orientable surface of genus g. Prove that M with a point removed has the same homotopy type as 2g circles with a point in common." I have ...
2
votes
1answer
56 views

Relative $CW$-complexes and Serre fibrations

Suppose we have a (continuous) map $p:E\rightarrow X$. Assume that $p$ has the right lifting property with respect to any relative $CW$-complex $i:A\rightarrow B$. I want to show that $p$ is a Serre ...
2
votes
1answer
76 views

An example of why this homomorphism of homotopy groups is not injective.

For a given path-connected space $X$, I recently learned that one could construct a CW complex $X_{1}$ by considering each generator $j: S^{q} \rightarrow X$ of $\pi_{q}(X)$ and setting ...
2
votes
1answer
39 views

Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
6
votes
1answer
97 views

Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$ \pi_k(A)=0=\pi_k(B) $$ for $k\neq n$. ...
1
vote
1answer
23 views

null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
0
votes
0answers
23 views

Basic question about abelianization of Homotopy Groups and Homology [duplicate]

When precisely, is the homology group: $$H_n(T)$$ of a topological space, $T$, isomorphic to the abelianiation of the corresponding homotopy group $\pi_n$? Does this only occur when $n=1$, or is ...
2
votes
0answers
47 views

mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
0
votes
0answers
57 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
2
votes
1answer
41 views

Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\} $ why it isn't homotopically equivalent to a CW-Complex?
1
vote
1answer
52 views

Show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$.

I'm studying for an oral qualifying exam in algebraic topology, going through questions in various tests published on the interwebs. Here's a rather straightforward question from this exam that is ...
2
votes
1answer
45 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
1
vote
1answer
49 views

Smash product and tensor product of groups

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). ...
1
vote
1answer
57 views

Attaching cells gives isomorphism of homotopy groups

I want to prove the following statement: Let $(X, x_0)$ be a pointed space, and let $X' = X\cup_{\alpha} e^{n+1}$ be obtained from $X$ by adjoining an $(n + 1)$-cell. Then the inclusion $i : ...
0
votes
0answers
23 views

Dual of path object

For a topological space X, what might be the dual of the path space $X^I$ of X? Does it make any sense to think of the topological cylinder X x I over X as dual to the path space over X?
2
votes
2answers
38 views

Fibration $p_1:P(Y,y_0)\to Y$ has section iff $Y$ is contractible.

Let $P(Y,y_0) = \{ \omega : \omega(0) = y_0 \}$ be path space let's consider a fibration $p_1:P(Y,y_0)\to Y$ such that $\omega \mapsto \omega(1)$. Show that there exists $s: Y \to P(Y,y_0)$ such ...
0
votes
1answer
33 views

Killing homotopy groups

I am basic with homotopy theory and especially with CW-approximation, Postnikov and Whitehead towers. In the proof of such things one need the following result on and on: Let $X'$ be obtained from ...
3
votes
0answers
38 views

Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...
0
votes
1answer
51 views

Homeomorphism “induced” by Lifting map. [closed]

I was going over this post :Explicit form of a lift $\tilde f: \tilde X_1 \to \tilde X_2$ of a continuous map $f: X_1 \to X_2$ $$\require{AMScd}\begin{CD} \tilde X_1 @>\tilde f >> \tilde ...