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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12
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247 views

Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
9
votes
0answers
421 views

Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
8
votes
0answers
272 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
7
votes
0answers
89 views

History of the term “anodyne” in homotopy theory

There is a notion of an anodyne morphism (usually of simplicial sets). This type of morphisms is used, for instance, to establish basic properties of quasicategories. But the name is quite mysterious. ...
7
votes
0answers
136 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
7
votes
0answers
134 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
6
votes
0answers
116 views

Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow ...
6
votes
0answers
49 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 $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
6
votes
0answers
338 views

Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
5
votes
0answers
71 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
5
votes
0answers
190 views

Homotopy equivalence in the category of arrows.

I'm reading Jeff Strom's book on Homotopy Theory and I am trying to make some sense of a certain exercise. On page 91, "Homotopy in Mapping Categories" we consider the category of arrows of ...
4
votes
0answers
23 views

Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$

When I think intuitively about homotopies, I think about them as paths between two functions. This is more comfortable and suggestive than any categorical talk about "morphisms between morphisms", so ...
4
votes
0answers
150 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
4
votes
0answers
96 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 ...
4
votes
0answers
77 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 ...
4
votes
0answers
45 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
4
votes
0answers
52 views

what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
4
votes
0answers
65 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
4
votes
0answers
63 views

Does an equivariant weak equivalence induce weak equivalences on all orbits?

This question arose from another, which was not well formulated and completely answered by this MO thread as pointed out by the user roman. Let $G$ be a discrete group, $X$ and $Y$ be $G$-spaces (and ...
4
votes
0answers
61 views

$J$-homomorphism and homotopy

We have Bott periodicity theorem for unitary group $U(n)$: $$ \pi_{i-1}^{s}(U) = \pi_{i-1}(U(m)) \simeq \pi_{i}(Gr_m(\mathbb{C}^{2m})) \simeq \pi_{i+1}(SU(2m)) \simeq \pi_{i+1}^{s}(U) .$$ So we can ...
4
votes
0answers
135 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
4
votes
0answers
74 views

Contractible and Compact space can be contained in an open set after time $t_0$?

$X$ is a topological space that is contractible and compact. Show that if $U$ is an open set in $X$ containing $x_0$ then there exists $t_0<1$ so that $H(x,t)∈U$, for all $x∈X$, and all $t_0≤t≤1$. ...
4
votes
0answers
108 views

A fibrant-objects structure on $\bf Top$

One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological spaces $\bf Top$, called the $\pi_0$-fibrant structure: A ...
4
votes
0answers
72 views

homotopy type of the closure of a subset

Let $X$ be a topological space and $N$ a subset of $X$. Is it true that the closure of $N$ in $X$ is homotopy equivalent to $N$. I think it is not. take for example $N=\mathbb Q\subset \mathbb ...
4
votes
0answers
112 views

Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?

This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly ...
4
votes
0answers
140 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
3
votes
0answers
77 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 ...
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 ...
3
votes
0answers
104 views

Maximum diameter for the preimage of a point when the degree is not 1 or -1

When the degree of a map $S^n \rightarrow S^n$ isn't 1 or -1, are there always two points that map to the same point and whose distance from each other can be bounded below by a function of the ...
3
votes
0answers
60 views

Definition of the triad homotopy groups

Let $ \ n \in \mathbb{N}$, $n \geqslant 2$. Equip $\mathbb{R}^n$ with the usual euclidean norm $ \ \Vert . \Vert : \mathbb{R}^n \to \mathbb{R}$, metric and topology. Consider $ \ p_0 = (1,0,0,...,0,0) ...
3
votes
0answers
60 views

About homotopy fiber at Hatcher's book

What is the meaning of the statement: In this case a map $(I^{i+1},∂I^{i+1},J^i) \to (B,A,x_0)$ is the same as a map $(I^i,∂I^i) \to (F_f, \gamma_0)$ where $\gamma_0$ is the constant path at ...
3
votes
0answers
39 views

Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?

In trying to remind myself why $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$, the natural argument that comes to mind is to take the long exact sequence associated to the fibration $S^1 \rightarrow S^\infty ...
3
votes
0answers
48 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
3
votes
0answers
40 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 ...
3
votes
0answers
29 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 ...
3
votes
0answers
50 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
3
votes
0answers
42 views

Derived pseudo-functor

Let $ \mathfrak {X}\to \mathfrak{Y} $ be a pseudofunctor (in which $\mathfrak{X} $ is a model category and $\mathfrak{Y} $ is a bicategory). I would like to understand when there is a derived functor ...
3
votes
0answers
85 views

Integral Homology of $BU$

We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$. And at almost ...
3
votes
0answers
73 views

Injective model structure

I equip the category of presheaves $[\mathcal{D}^{op},\text{Gpd}]$ with the injective model structure ($\mathcal{D}$ is just any small category). In this structure, weak equivalences and cofibrations ...
3
votes
0answers
157 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
3
votes
0answers
230 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
3
votes
0answers
126 views

Homotopy of a CW complex

I have a CW complex constructed as follows: (The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein ...
3
votes
0answers
61 views

Does the $\mathcal{E}G$-construction arise form a comonad?

Let $G$ be a simplicial group and consider the adjunction $$ Fr:sSets\rightleftarrows G-sSets:U $$ where the left adjoint $Fr$ maps a simplicial set $X$ to the free $G$-simplicial set $G\times X$ and ...
3
votes
0answers
106 views

simple proposition about homotopy group

Let $A$ be a topological subspace of $X$. Then we have exact sequences of homotopy groups for pair spaces: $$ \ldots \rightarrow \pi_n(A,x_0) \xrightarrow{i_{*}} \pi_n(X,x_0) \xrightarrow{j_{*}} ...
3
votes
0answers
397 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
3
votes
0answers
75 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
3
votes
0answers
71 views

Is a simplicial subcomplex of a contractible simplicial CW-complex also contractible?

Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy ...
3
votes
0answers
66 views

Removing the star without changing homology

I know that if the link of a simplex $\sigma$ in a finite simplicial complex $K$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type. ...
3
votes
0answers
50 views

How to show $\mathbb{S}^{2n-1}$ is a $\mathbb{S}^{n-1}$ bundle over $\mathbb{S}^{n}$ then $n=1,2$ or divisible by $4$?

I was asked to show if $$\mathbb{S}^{n-1}\rightarrow \mathbb{S}^{2n-1}\rightarrow \mathbb{S}^{n}$$ holds. Then $n=1,2$ or divisible by $4$. The $n=1,2$ cases in the converse are verfied. I am ...
3
votes
0answers
42 views

some help on the group of unknotted

Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$. ...