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37
votes
8answers
3k views

What are the uses of the homotopy groups of spheres?

Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?: Have the homotopy groups of spheres ever been applied to ...
23
votes
1answer
2k views

What is so “spectral” about spectra?

What is the background of the terminology of spectra in homotopy theory? In what extend does the name "spectrum" fit to the definition and the properties? Also, are there relations to other spectra in ...
6
votes
1answer
785 views

Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander

Let $X_\bullet \longrightarrow Y_\bullet \longleftarrow Z_\bullet$ be a diagram of simplicial spaces (=bisimplicial sets, if you like). On p. 14-9 of these notes there is an example which shows that ...
3
votes
2answers
438 views

A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
6
votes
1answer
545 views

A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a regular CW complex of constant local dimension $n$. $X_{n}$ is of finite type, ...
8
votes
4answers
1k views

Whitehead for maps

I made the following claim over at the Secret Blogging Seminar, and now I'm not sure it's true: Let f: X --> Y and g: X --> Y be two maps betwen finite CW complexes. If f and g induce the same map on ...
6
votes
2answers
463 views

Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying : $X_{n}$ have topological dimension $n$. $X_{n+1}$ is ...
4
votes
3answers
472 views

Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below : This post has been divided into two parts, the second part is here. Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
4
votes
1answer
414 views

Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

Motivation In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in ...
58
votes
6answers
3k views

What properties make $[0,1]$ a good candidate for defining fundamental groups?

The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
24
votes
4answers
3k views

What is the “intuition” behind “brave new algebra”?

Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
31
votes
9answers
2k views

Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
18
votes
6answers
2k views

Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
28
votes
2answers
2k views

Are there pairs of highly connected finite CW-complexes with the same homotopy groups?

Fix an integer n. Can you find two finite CW-complexes X and Y which * are both n connected, * are not homotopy equivalent, yet * $\pi_q X \approx \pi_q Y$ for all $q$. In Are there two ...
24
votes
3answers
3k views

Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
15
votes
5answers
2k views

A homotopy commutative diagram that cannot be strictified

By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to ...
15
votes
5answers
2k views

Computing homotopies

Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to ...
12
votes
3answers
1k views

What determines a model structure?

It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal). cofibrations and weak ...
20
votes
3answers
1k views

Modern Source for Spectra (including Ring Spectra)

I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been ...
15
votes
4answers
1k views

Classifying Space of a Group Extension

Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example: $$ 0 \to H \to G \to G/H \to 0\ . $$ I want to understand the classifying space of $G$. Since ...
15
votes
3answers
2k views

Different way to view action of fundamental group on higher homotopy groups

There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
13
votes
8answers
2k views

References for homotopy colimit

(1) What are some good references for homotopy colimits? (2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
19
votes
0answers
485 views

Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

For a (discrete) monoid $M$, the classifying space $BM$ is the geometric realization of the nerve of the one object category whose hom-set is $M$. (This definition gives the usual classfiying space ...
14
votes
3answers
2k views

Conjectures in Grothendieck's “Pursuing stacks”

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this ...
10
votes
2answers
889 views

homotopy type of embeddings versus diffeomorphisms

Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question ...
9
votes
3answers
660 views

What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
8
votes
5answers
2k views

Are there two non-homotopy equivalent spaces with equal homotopy groups?

Could someone show an example of two spaces X and Y which are not of the same homotopy type, but nevertheless \pi_q(X)=\pi_q(Y) for every q? Is there an example in the CW complex or smooth category?
15
votes
1answer
388 views

Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
33
votes
0answers
728 views

Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...
7
votes
2answers
325 views

What is a Homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
5
votes
3answers
740 views

categorical homotopy colimits

let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
7
votes
1answer
365 views

Three questions on $\operatorname{hocolim}$

I posted this on math.stack.exchange but didn't get a helpful response, so please let me try it here. Let $D$ be a small category and $F:D\to sSets$ a functor. There is a bisimplicial set indicated ...
4
votes
1answer
530 views

references for models of stable infinity categories

There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
20
votes
1answer
449 views

Is $\mathbb{H}P^\infty_{(p)}$ an H-space?

Put $X=\mathbb{H}P^\infty$ (so $X$ classifies quaternionic line bundles, and $\Omega X=S^3$). There is no obvious reason for $X$ to be an H-space, because the tensor product of quaternionic vector ...
11
votes
2answers
435 views

Noncontractible connected topological rings ?

Are there any non-contractible connected topological rings? Of course, such a thing cannot be a (topological) algebra over the reals. (I have a vague memory of having a glance at an erticle by ...
9
votes
2answers
523 views

Configuration spaces and non homeomorphic vector bundles

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign. ...
8
votes
4answers
621 views

The most general context of Mather's Cube Theorems

Quite simply, I'd like to know what is the broadest or most natural context in which either (or both) of Mather's cube theorems hold. If you like, this may mean any of What properties of $Top$ or ...
19
votes
2answers
1k views

How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context. Let $\mathcal{C}$ be a category. ...
19
votes
3answers
595 views

Is the counit of geometric realization a Serre fibration?

Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the ...
10
votes
1answer
771 views

The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
8
votes
1answer
298 views

Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
4
votes
1answer
340 views

Homotopy of quivers

The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...
4
votes
1answer
306 views

Finite homotopy limits commute with sequential homotopy colimits

I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems ...
9
votes
2answers
1k views

Well-pointed space which is not locally contractible

I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
6
votes
2answers
384 views

mapping space between classifying spaces

I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces ...
5
votes
0answers
198 views

Mod 3 Moore spectrum

I only know through stories that mod 3 moore spectrum is not associative. I do not know of any proof. I have been informed that Toda had proved it in the paper "Extended $p^{th}$ power". I was not ...
4
votes
2answers
735 views

Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...
2
votes
0answers
190 views

A topos-theoretic thickening of the nerve functor

Let $\Delta$ be the standard cosimplicial space sending $[n]\mapsto \Delta^n\subset \mathbb R^{n+1}$. Then the correspondence $\delta\colon [n]\mapsto {\rm Sh}(\Delta^n)$ defines a cosimplicial ...
1
vote
1answer
196 views

Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
0
votes
0answers
62 views

Problem on infinite dimensional metric space, with rigidity assumption

By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : ...