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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2
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1answer
30 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
2
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0answers
39 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
1
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0answers
13 views

Homotopy for the complement when slightly thickening a subspace of $ℝ^n$

For a given subspace $X \subset ℝ^n$ there is a homotopy equivalence $X \times \{0\} \simeq X \times I$ (subspaces of $ℝ^{n+1}$), where $I$ is the unit interval. However, the complements in $ℝ^{n+1}$ ...
6
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2answers
101 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
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0answers
40 views

Homotopy groups of unitary groups

in this paper I found some explicit generators of homotopy groups of unitary groups, for example $\pi_3[SU_2]$: $\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 ...
1
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0answers
33 views

$\pi_0$ of $M(2) \wedge M(2)$

My motivation is trying to understand Tom Goodwillie's argument here: http://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why ...
2
votes
2answers
35 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
3
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3answers
51 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...
6
votes
1answer
136 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
2
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0answers
23 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
1
vote
1answer
22 views

Being smooth homotopic relation: proof

Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that ...
1
vote
1answer
100 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
1
vote
1answer
23 views

What is the meaning of “Homotopy of Little disc Operads”

I want to understand what means the homotopy of the little discs operad. I'm starting to research in this area and I have some questions. 1) I don't understand why little discs operad is a ...
3
votes
2answers
75 views

Topological/homotopical classification for 1-dim CW-complexes?

It's a common exercise to classify a collection of 1-dim objects, say the figures of 0-9, or A-Z, up to homeomorphism or homotopy equivalence. I suddenly raise a question in general: Is there any ...
2
votes
0answers
26 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 ...
1
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0answers
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
120 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
29 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
votes
1answer
99 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
votes
0answers
52 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 ...
1
vote
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
votes
2answers
149 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
votes
1answer
63 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
29 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
votes
1answer
50 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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0answers
22 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.
0
votes
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 ...
2
votes
2answers
74 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
votes
1answer
59 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 ...
0
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0answers
31 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 ...
0
votes
0answers
26 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?
1
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0answers
28 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
88 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
30 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
vote
1answer
27 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
vote
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
votes
1answer
38 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 ...
1
vote
0answers
58 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 ...
1
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0answers
50 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
57 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
87 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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vote
0answers
27 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
25 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 ...
1
vote
0answers
44 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
53 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+ ...