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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14
votes
1answer
349 views

Is there a generalization of homotopy groups to fractional dimensions

Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?
7
votes
1answer
352 views

Loop space generalization

Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$ is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space ...
1
vote
0answers
363 views

Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?
13
votes
0answers
567 views

Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question. In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...
-5
votes
0answers
65 views

A question on algebraic topology [closed]

Show that there is no one to one continuous map R^n to R^2 for n > 2 with f(0) =0. I try to solve by assume if there is no such continous exist but couldnot get way to go further. Any help is really ...
0
votes
1answer
290 views

Dimension of two homotopy equivalent manifolds [on hold]

Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of ...
3
votes
0answers
212 views

Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
0
votes
0answers
72 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
2
votes
0answers
60 views

section spaces related to configuration spaces

In the paper Configuration spaces of positive and negative particles, Dusa McDuff, a section space $\Gamma(M)$ is constructed: And in the paper ON THE HOMOLOGY OF CONFIGURATION SPACES. C.-F. ...
2
votes
0answers
90 views

homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$ $$ ...
4
votes
1answer
182 views

Properties of loop space functor from homotopy types to group objects in homotopy types

I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important: When we take the loop-space of a (connected) homotopy type, we get ...
4
votes
1answer
135 views

Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where ...
4
votes
1answer
170 views

Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...
3
votes
0answers
59 views

cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any ...
1
vote
1answer
45 views

Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$ \alpha_n: C(\mathbb{R}^n;X)\to ...
3
votes
0answers
75 views

Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...
2
votes
1answer
143 views

Equivalent definition of a Kan fibration

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect ...
1
vote
1answer
98 views

maps from labelled configuration space to section space / iterated loop space

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3: for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle ...
8
votes
2answers
398 views

Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...
1
vote
0answers
47 views

cohomology ring of unordered configuration space on Euclidean spaces

Let $F(\mathbb{R}^n,k)/\Sigma_k$ be the $k$-th unordered configuration space on $\mathbb{R}^n$. In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 226, it ...
12
votes
3answers
1k views

$(\infty,1)$-categories and model categories

I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can one associate an ...
6
votes
1answer
186 views

Infinity category of functors from a relative category to a model category

Let $M$ be a model category (maybe cofibrantly generated/combinatorial). Let $(C,W)$ be a relative category. I write $M^{(C,W)}$ for the full subcategory of $M^C$ on relative functors. This is a ...
4
votes
1answer
257 views

Natural transformations induce homotopies - Is this true in the “fat” world?

Let $\mathcal{C}, \mathcal{D}$ be categories internal to topological spaces (or compactly generated Hausdorff spaces, if you like) $F,G\colon\mathcal{C}\rightarrow\mathcal{D}$ be continuous functors ...
4
votes
0answers
130 views

Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

I'm willing to work in the category of compactly generated Hausdorff spaces. The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...
22
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 ...
7
votes
0answers
342 views

What's the detailed proof of “the composition of planar tangles is well-defined”?

In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...
13
votes
2answers
697 views

Counterexamples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...
3
votes
1answer
392 views

$\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened ...
10
votes
3answers
938 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 ...
3
votes
1answer
113 views

group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
6
votes
2answers
164 views

Direct proof that the model category of cdgas is left proper

Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
4
votes
1answer
294 views

$\Omega X$-action on spectral $X$-bundles

I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the ...
5
votes
0answers
146 views

Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map $$ S^0 \overset{p^i}\longrightarrow S^0 $$ where $S^0$ be the sphere spectrum. In the Mathoverflow ...
6
votes
2answers
343 views

On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
1
vote
0answers
66 views

Internal Hom on simplicial presheaves and the preservation of cofibrant objects

1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, ...
6
votes
3answers
470 views

Higher refinement of Seifert-van Kampen theorem on the language of hocolim

I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then $$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: ...
9
votes
2answers
662 views

Homology of localisations of spectra

Let $H^*$ and $K^*$ be two cohomology theories, and $X$ a reasonable spectrum. Here, I'm thinking that $H^*$ is singular cohomology (and for my purposes, rational cohomology will suffice), and $K$ is ...
8
votes
0answers
170 views

Simplices and cubes

Question: What is the first appearance in the literature of one of the following statements: The result of intersecting a simplex with a cell of the dual subdivision is a cube There ...
3
votes
1answer
169 views

Does the CGWH-fication change the (weak) homotopy type?

Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces. There is the CG-ification ...
3
votes
0answers
63 views

Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer: Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...
14
votes
9answers
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 ...
9
votes
1answer
212 views

Tensor products over operads and bar constructions

Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in ...
40
votes
6answers
3k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
0
votes
3answers
79 views

smash product of pointed spaces preserve weak equivalences

Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor $$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does ...
-2
votes
1answer
118 views

configuration space and iterated loop space

Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...
7
votes
2answers
512 views

Embedded (framed) cobordisms

[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.] This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...
4
votes
1answer
262 views

localization and $E_{\infty}$-spaces

Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff ...
1
vote
0answers
100 views

group completion of topological monoid

Let $M$ be a topological monoid. The group completion of $M$ is defined as $\Omega BM$, where $BM$ is the classifying space of $M$. In Lecture Notes in Math. 533, The homology of C n+1 spaces, n>=0, ...
2
votes
0answers
95 views

Nilpotency of BG^+

I have become interested in knowing if there are conditions, in the literature, under which the action of the fundamental group of BG^+ acts nilpotently on its higher homotopy groups, where the + ...
3
votes
1answer
85 views

Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...