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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28 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
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27 views

Showing this Null homotopic composite factors through a Null homotopic map

I was having some trouble with this concept which makes sense to me intuitively but the understanding of which is not yet fully clear. Suppose $CS^n$ is the unreduced cone on the n-sphere $S^n$. By ...
4
votes
3answers
127 views

Do freely homotopic maps induce the same homomorphism on fundamental groups?

Let $f,g\colon X\to Y$ be two continuous maps that are freely homotopic, such that there is some $x_0\in X$ with $f(x_0)=g(x_0)$. Is it true that the induced homomorphisms $f_*,g_*\colon ...
1
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0answers
37 views

Proof the Associative Property of H-group

I would like to prove the following proposition: Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a ...
2
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1answer
111 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
4
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0answers
53 views

Contractible Subspace and Homotopy Equivalence

It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence. I am wondering what if the homotopy ...
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2answers
73 views

Homotopy equivalent iff isomorphic homotopy groups?

Is it true that two spaces or $\infty$-groupoids are homotopy equivalent if and only if they have isomorphic homotopy groups?
6
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2answers
154 views

Can we simultaneously realize arbitrary homotopy groups and arbitrary homology groups?

Let's keep our groups finitely presented for the time being. All spaces in this post are path connected. Background: By a standard construction (e.g., on p. 365 of Hatcher), there exists a $K(\pi, ...
4
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1answer
68 views

Homological algebra (homotopical approach)

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is ...
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0answers
31 views

Showing that a homotopy fiber of a fibration is homotopy equivalent to the fiber of the base point.

Assume $(E, e_0)$ and $(B, b_0)$ are based spaces with the indicated base points. Given a based fibration $p: E \rightarrow B$. We have the respective homotopy: fiber \begin{equation} Fp= ...
2
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1answer
53 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
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26 views

Reference for introductory text on Homotopy Theory

Can anyone recommend a good introductory text on Homotopy Theory? Paid textbooks or free online material/lecture notes both welcome.
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1answer
50 views

A problem concern about the relationship between homotopy groups.

I encountered the following problem: Let $X$ be a closed manifold, $S^n$ be the $n$-dimensional standard sphere and denote $\Omega(S^n,X)$ be the space of base point-preserving maps from $S^n$ to ...
3
votes
2answers
77 views

What's the calculation formula of topological number for mappings of $\pi_{3}(S^2)=\mathbb{Z}$?

It is well-known that, when mapping $|\vec{n}(\vec{x})|=1$, we can use $N=\int{\mathrm{d}x_1\mathrm{d}x_2\vec{n}\cdot(\partial_1\vec{n}\times\partial_2\vec{n})}$ to calculate the topological winding ...
2
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1answer
65 views

Obstruction to reduction of structure group

In the wiki article it's stated that the obstruction to reduction of structure group along a morphsim $H \to G$ can be stated in terms of classifying spaces via the cofibre $BG/BH$ as follows. A ...
0
votes
1answer
38 views

About the Degree of a Map

I am reading Elements of Homotopy Theory by George W. Whitehead. In the section about maps of the $n$-sphere into itself, in the second last paragraph of the text quoted below, he says that "Then an ...
1
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0answers
37 views

Second Volume of Elements of Homotopy Theory?

In the preface of Elements of Homotopy Theory (GTM 61) by George W. Whitehead, he wrote that "I plan to devote a second volume to these developments". Does any one know if George eventually published ...
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0answers
27 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
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0answers
36 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
3
votes
3answers
92 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
0
votes
1answer
32 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
1
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1answer
28 views

Proving the left lifting property for a map

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence. I don't know how to draw a square ...
1
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1answer
24 views

In the definition of $n$-equivalence, what is the motivation for only requiring surjectivity on the $n$th dimension.

An $n$ equivalence $f\colon X \to Y$ such that the induced map on the homotopy group $f_* \colon\pi_m(X) \to \pi_m(Y)$ is an isomorphism for $m<n$ and an epimorphism for $m=n$. What's the ...
0
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1answer
39 views

The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent.

I will appreciate if someone could explain to me the solution of the following problem: The unit ball in $\mathbb{R}^{n}$ and a point are homotopically equivalent. Def 1: Two spaces $X$, $Y$ are ...
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0answers
77 views

Cofiber Sequences in Reduced Homology Theory

While going through axiomatic treatments of homology theories I got a bit stuck on this problem. Consider given a reduced homology theory, i.e. functors $(\tilde{E}_q:Top_* \to Ab)_{q \in ...
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0answers
54 views

morphisms of principal bundles with different structure groups

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram: \begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G ...
3
votes
2answers
60 views

Show $f$ has a fixed point if $f\simeq c$

I have the following problem: Show that if $f:S^1\to S^1$ is a continuous map, and $f$ is homotopic to a constant, then $\exists p\in S^1 : f(p)=p$. My approach is to show that if for all $p, \ $ ...
8
votes
1answer
90 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
2
votes
1answer
54 views

Showing $S^1$ is not a retract of $D^2$ using homotopy

I'd like to know if my argument below, in which I try to show that the 1-sphere is not a retract of the 2-disk using homotopy, is valid. Suppose there is a retract $r:D^2 \to S^1$. Then we can ...
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0answers
28 views

Stack on commutative ring spectra?

One approach to stacks to call a stack a "sheaf of groupoids" which means a functor $$ \mathcal{C}^{\text{op}} \rightarrow \mathcal{G} $$ from a category $\mathcal{C}$ with a Grothendieck topology to ...
3
votes
0answers
28 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
2
votes
1answer
48 views

Use Hurewicz Theorem to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$

Want to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$, using Hurewicz theorem. This is one of the questions on the previous topology qualifying exams. Any help will be appreciated! I am thinking in stead ...
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0answers
45 views

Why the Objects of Homotopy Category not Homotopy Classes of Spaces?

A homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. I wonder why the objects are spaces, instead of homotopy ...
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0answers
59 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
2
votes
2answers
32 views

Prove sequence has limit in $\gamma (S^1)$

This is a seemingly interesting exercise from my topology notes, but I can't solve it for the life of me. It's like this: take a closed curve $\gamma : S^1\to \mathbb R^2$, and a sequence ...
0
votes
1answer
27 views

Example of non-homotopic functions

I start to study homotopy theory and I've trouble with following "proof": "If $f_0$, $f_1$ are continous functions $X \rightarrow Y$, then one can consider the continous function $F=(1-t)\cdot f_0+ ...
1
vote
1answer
41 views

Does ''homology vanishes eventually'' imply ''homotopy vanishes eventually''?

Let $X$ be a connected CW complex. Assume there is an integer $N\geq 0$ such that the singular homology $H_n(X)=0$ vanishes for all $n\geq N$. Is there an integer $M\geq 0$ such that $\pi_m(X)=0$ ...
0
votes
0answers
15 views

Does the homotopy cofiber detect obstruction to lifting?

Let $f:A\to X$ be a based cofibration with $C$ as homotopy cofiber. Let $g: T\to X$ be a based map. If $g$ lifts to $A$, that is to say, there exists a based map $\tilde{g}:T\to A$ such that $f\circ ...
2
votes
0answers
31 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
1
vote
2answers
61 views

Homotopy equivalence between $X/A$ and $X$?

Consider the following definition: Definition: Let $(X, A)$ be a topological pair. We say $A$ has the homotopy extension property with respect to a space $Y$ if given any continuous map ...
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votes
0answers
34 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
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0answers
72 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
5
votes
1answer
80 views

Why are the total spaces of two Serre fibrations equivalent when the bases and the fibers are equivalent?

Suppose $B$ is a pointed space and suppose $f\colon E\to B$ and $f\colon E'\to B$ are two Serre fibrations. Let moreover a map $g\colon E\to E'$ be given such that $f=f'\circ g$ which is a weak ...
2
votes
0answers
30 views

Contractible as an Unbased Space but Not Contractible as a Based Space

An unbased space $X$ is contractible if $id_X$ is homotopic to a constant function, that is, any function which carries all of $X$ to single point. Is there an unbased contractible space $X$ such ...
0
votes
1answer
46 views

Contractible vs. Contractible in a space

I am reading Introduction to Homotopy Theory by Arkowitz Martin and on page 9 it reads: More generally, if $A$ is a subset of $X$ with inclusion map $i : A \to X$; then $A$ is contractible in $X$ ...
4
votes
1answer
170 views

Is a Simply Connected Space Homotopically equivalent to a point

If X is a simply connected Space is it homotopically equivalent to a point in the space, I know this holds in $ \mathbb{R}^n $ but only because of its algebraic properties does it hold for general ...
0
votes
0answers
13 views

What is the homotopy type of the orbit space of the conjugation involution on complex projective space?

The generator of the cyclic group of order 2 acts on 2-dimensional complex projective space $\mathbb{C}P^2$ by sending a point with homogeneous coordinate $[Z_0: Z_1 : Z_3]$ to its conjugate point $[ ...
4
votes
1answer
56 views

Topology of space of symmetric matrices with fixed number of positive and negative eigenvalues

Let $M$ be real non-singular symmetric $n \times n$ matrix with $p$ positive and $n-p$ negative eigenvalues. What is the topology of the space of such matrices? For a trivial case $n=1$ the matrix is ...
1
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1answer
33 views

homotopical equivalence of projective real space less a line

let $r$ a projective line of the projective real space. How can i prove that $\mathbb{P} ^3(\mathbb{R}) - r$ is homotopical equivalent to $S^1$?
2
votes
2answers
51 views

Foundamental group of $n+1$ spheres in $\mathbb{R}^{n+1}$ that touch two by two

How can i calculate the foundamental group of three $S^2$ in $\mathbb{R}^3$ that touches two by two in one point (if you take any two spheres, they touch only in one point) ? I know that is ...