Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas ...

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276 views

Homotopy groups of Moore spaces

Is there anything known about the homotopy groups of the Moore spaces $M(\mathbb Z_m,n)$ if $m\neq 2$ and $n \geq 2$?
1
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1answer
155 views

Free Symmetric Operads and $\mathbb{S}$-modules

In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads have still an underling free ...
3
votes
2answers
98 views

Cube Lemma on a cofibrantly generated (almost) model category

Suppose I have a complete and cocomplete category $\mathscr{C}$ with two sets of maps $I,J$ that are the candidates for generating (trivial) cofibrations on a model structure on $\mathscr{C}$. The ...
0
votes
0answers
46 views

Small perturb a continuous map

I am reading the book: Convex Integration Theory by D. Spring and encounter a question which I subtract as follows. Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset ...
19
votes
2answers
819 views

The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts. What ...
2
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0answers
73 views

Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group $G$, and we pick two elements $g,h \in G$, could we find the order of the element $g \wedge h \in G \wedge G$? The best thing I could find is Theorem 1.1 in Ellis' Book ...
3
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0answers
88 views

Universal enveloping algebra functor preserves quasi-isomorphism

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model ...
5
votes
1answer
327 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
7
votes
2answers
351 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
10
votes
2answers
463 views

Reference request: Goodwillie tower of the identity

The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor $$ X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]}) $$ in ...
8
votes
2answers
339 views

Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids

Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations. Generally $Vect(BG) \ne ...
2
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0answers
195 views

E-infinity operads explicit examples

I was looking for particular and explicit examples of $E_\infty$-operads. I know the $E_\infty$-operad defined by Smith in http://arxiv.org/abs/math/0004003, and the Barratt-Eccles operad, but it is ...
4
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0answers
91 views

Cofibrations in the model structures for non-negative graded (commutative) DG algebras

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...
5
votes
0answers
95 views

Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle). I ...
8
votes
0answers
157 views

Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
0
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0answers
41 views

Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property. Here is the definition of f.p.s.p.("map" means ...
8
votes
1answer
153 views

Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
5
votes
1answer
145 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...
15
votes
2answers
349 views

Table of (integral) cohomology groups of K(Z,n)

Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients? It seems like a natural counterpart to the table of the homotopy groups of spheres, ...
1
vote
0answers
116 views

Is every path connected space continuously path connected

Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$. Say that $X$ is continuously path ...
7
votes
1answer
89 views

Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories $$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$ containing all the isomorphisms, such that the following ...
10
votes
1answer
255 views

Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...
3
votes
0answers
152 views

Complete Segal operads and dendroidal sets

There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the ...
4
votes
1answer
151 views

Formula relating the cup product in dimensions n and n+1

Let's will write $K_n$ for the Eilenberg-MacLane space $K(\mathbb{Z},n)$. I remind that $K_n$ is equivalent to the loop space of $K_{n+1}$. Let’s consider the map $\smallsmile:K_n\times K_m \to ...
23
votes
2answers
869 views

What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
3
votes
1answer
202 views

(Geometric) Proof for the projective bundle formula in K-theory

I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove: Projective bundle formula: Let $\pi: E \to X$ be a ...
6
votes
1answer
317 views

Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?

When reading about homotopy algebras (e.g. $L_\infty$-algebras, $A_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f_1$ is a quasi-isomorphism. Recall/Example ...
12
votes
2answers
372 views

H-space structures on non-sphere suspensions?

It is well known that $S^n$ admits an H-space structure if and only if $n=0,1,3,7$. I'm interested in whether there are other suspensions $\Sigma X$ that admit H-space structures: Question 1 For ...
13
votes
2answers
418 views

Discrete Morse theory: how do zig-zag paths determine homotopy type?

Let $\Delta$ be a simplicial complex (or more generally, a regular CW complex). Let $\mathcal{M}$ be a Morse matching (or equivalently, a discrete Morse function) on $\Delta$. By Forman's theorems, ...
12
votes
1answer
230 views

From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two ...
7
votes
1answer
236 views

Closed formulas for topological K-theory?

Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line ...
3
votes
0answers
141 views

The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
5
votes
0answers
170 views

Homotopy groups of the Grassmannians

What are the homotopy groups of the oriented Grassmannian $Gr^{+}(p,q)$ (p-planes in $R^{p+q}$) $\pi_{r}(Gr^{+}(p,q))$, $r \le pq$? Do you know any references on the web about it?
13
votes
2answers
379 views

When do colimits agree with homotopy colimits?

I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F_c)$ where $F_c$ is the cofibrant replacement of $F$. However, it ...
13
votes
1answer
324 views

Homotopy fiber of a map between classifying spaces

I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
14
votes
2answers
420 views

$RO(G)$-graded homotopy groups vs. Mackey functors

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically. Also, I've looked through other similar MO questions, but I didn't find ...
8
votes
1answer
230 views

Relative version of Quillen's theorem A

Quillen's Theorem A is formulated as follows: Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category $F/y$ is contractible. Then $F$ induces a weak equivalence ...
3
votes
0answers
65 views

SImple homotopy type of a mapping cone

Consider two CW complexes A and B. Let f be a continuous map from A to B. Take a cone of f and denote it by Cone(f). If homology complex of Cone(f) is acyclic, one can identify homologies of A and B. ...
47
votes
2answers
2k views

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
13
votes
0answers
394 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ...
5
votes
0answers
163 views

What makes Reedy model categories useful?

I have been reading up a bit on the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$ in Goerss-Jardine. One thing I find a bit unclear is what the Reedy ...
1
vote
1answer
122 views

pullback square in Goerss-Jardine

In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of ...
15
votes
0answers
271 views

Does the “holomorphic spheres-to-continuous spheres” forgetful function respect the mixed Hodge structures on homotopy groups?

For each smooth, projective, complex variety $X$ that is simply connected, John Morgan constructed a natural mixed Hodge structure on the homotopy group $\pi_k(X,x)\otimes \mathbb{Q}$. This was ...
11
votes
1answer
252 views

Whitehead products and Framed Manifolds

The attaching map for the top cell of the torus $S^n \times S^n$ is a map $$ [x,y]: S^{2n-1} \to S^n \vee S^n $$ where the notation is such that $x,y : S^n \to S^n \vee S^n$ are the two ...
8
votes
1answer
248 views

Can the groupoid completion of a topological category be recovered from its classifying space?

Let $C$ be a category. The groupoid completion of $C$ is the free groupoid on $C$, i.e. the category $C[C^{-1}]$ obtained by localizing at everything. Recall that the classifying space $\mathbf{B}C$ ...
5
votes
0answers
130 views

Push forward of the constant sheaf for a Serre's fibration

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
18
votes
1answer
648 views

How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
8
votes
1answer
185 views

Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
4
votes
0answers
62 views

Homology of simplicial manifolds

Let $M_{\bullet}$ be a simplicial manifold. There are two ways to computing its cohomology. Consider the cosimplicial module $A_{DR}(M)$. It defines a functor $A_{DR}(M_{\bullet})\: : \: ...
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0answers
112 views

Structures on the free path space object

Let $X$ be a connected topological space. Then its based loops space $\Omega X$ is an examples of $A_{\infty}$ space. Let $PX$ be the free paths space. Which type of homotopical structures are ...