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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11
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419 views

What if homotopy were expanded to allow any connected space instead of [0,1]?

What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than [0,1]? Given continuous $f,g:X\to Y$, define $f$ and $g$ to be ...
5
votes
1answer
110 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 ...
9
votes
1answer
167 views

Dyer-Lashof operations and the homology of GL_n

For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...
1
vote
1answer
151 views

Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...
21
votes
2answers
924 views

Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather ...
4
votes
1answer
219 views

When are principal bundles preserved by colimits?

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...
3
votes
2answers
230 views

Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...
4
votes
2answers
280 views

Maps to the group completion

Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary): What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...
2
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0answers
127 views

Groups acting on complexes

Let $G$ be a finite group. We define a $G$-simplicial complex $\mathcal{A}(G)$ with set of vertices $G^*:=G-\{e\}$ and the simplices are the abelian subsets of $G^*$. The groupe $G$ act simplicially ...
10
votes
1answer
272 views

Cohomology of the Image of J spectrum

Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e. $$ [j, HZ/p]_*$$ as a module over Steenrod algebra. ...
4
votes
1answer
172 views

Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions. Is the infinity-groupoid of a finite ...
5
votes
1answer
168 views

Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following. There exist $\Sigma$-free operads ...
1
vote
0answers
89 views

Explicit calculation of G-CW(V) structure of a G-space

I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...
1
vote
1answer
89 views

Homotopy invariance of Kan nerve of simplicial categories

The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the ...
1
vote
0answers
29 views

Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR

Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$? ...
4
votes
0answers
241 views

A construction with homotopy colimits and homotopy pullbacks for descent

EDIT: Following the lines of some suggestions in the comments below, I try to add something more to explain the problem better. A map $\text{hocolim}Y\rightarrow\bar{Y}$ in $\text{Ho}(\mathbf{M})$ is ...
3
votes
1answer
180 views

Glueing a property via homotopy colimits

I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf . In the proof of Lemma 2.11, we are given a ...
3
votes
0answers
81 views

Homotopical categories, the 2-out-of-6 property, and saturation

A homotopical category is a cateogry with a class $\mathcal W$ of arrows called weak equivalences which satisfies the 2-out-of-6 property. The nlab article shows a deep connection between $\mathcal ...
5
votes
1answer
171 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

I asked this question on math.stackexchange a week ago, but did not get an answer. First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...
5
votes
1answer
228 views

When does the Borel construction have the homotopy type of a CW-complex?

Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? ...
1
vote
0answers
79 views

Why is the oriented $G$-homotopy type of a $G$-complex uniquely determined by the periodicity generator?

Say we have a periodicity generator $e \in H^k(BG)$. I can show that we then have a $(k-1)$-dimensional $G$-complex $X$ with free $G$-action. It's also not that difficult to see that it has trivial ...
8
votes
3answers
427 views

When are (weak) homotopy equivalence testable on open covers?

I asked this question on math.stackexchange, but did not get an answer. Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't ...
2
votes
1answer
160 views

cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...
5
votes
2answers
216 views

homology of a mapping spectrum

If $X$ and $Y$ are two spectra, I denote by $F(X,Y)$ their mapping spectrum. This is uniquely determined by the existence of a natural isomorphism $[X\wedge Y, Z]\cong [X,F(Y,Z)]$. I denote by $H_*$ ...
0
votes
0answers
19 views

Criterion for projectively cofibrant diagrams? [duplicate]

(repost from math.stackexchange) Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The ...
23
votes
1answer
646 views

Homotopy Type Theory: What is it?

My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following: There are three directions: ...
10
votes
1answer
194 views

Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?

There are twelve continuous maps $S^3\times S^3\to S^3$ up to homotopy that make the three-sphere $S^3$ into an H-space. This follows from a result of James [1], which says that if there exists one ...
0
votes
0answers
50 views

question about Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces

In the paper Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, F. R. Cohen, L. R. Taylor, Geometric Applications of Homotopy ...
0
votes
0answers
64 views

Dyer-Lashof algebra structures over graded modules

In Lecture Notes in Mathematics, Vol. 533, The homology of iterated loop spaces, Chapter 3, The homology of $C_{n+1}$-spaces, F. Cohen, Section 2, page 222, line 4, 5, 6: for an arbitrary graded ...
3
votes
1answer
235 views

Dimension leaking in homology as opposed to homotopy

In homotopy theory we have the Seifert van-Kampen theorem, which is a clean statement about the fundamental groupoid of a pushout in $\mathsf{Top}$. There is also a 2d version of SvK in R Brown's ...
5
votes
2answers
282 views

Choice of fibrations is like a choice of a basis of a module

In some notes on derived stacks, in describing categories of fibrant objects, the author drops this parenthetical: (Grothendieck said in his famous letter to Quillen that the choice of $\mathscr ...
6
votes
1answer
262 views

Properties of coefficients of ring spectra

This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law ...
1
vote
0answers
154 views

Equivariant Homotopy

Let $G=\mathbb{Z}/2\mathbb{Z}$ be $\{\pm1\}$ and let there be two $G$-spaces given: $X=$ The surface of a cylinder including its boundary circles and $S^4$. That means we two G-actions $f_1:G\times ...
12
votes
1answer
377 views

Lemma 2.1.1.4 in Lurie's HTT

I have encountered a problem in understanding Lurie's proof of the following fact: "Given a left fibration between simplicial sets $q:X \to S$, there exists a functor $$ho(S) \to Ho(sSet)$$ which is ...
8
votes
4answers
734 views

Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...
0
votes
1answer
69 views

Homotopy bounds in simply connected complete Riemannian manifolds

Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all paths ...
6
votes
1answer
259 views

Explict form of $E_\infty$-morphisms between differential graded commutative algebras

This is a partial duplicate to this MO question, I apologize for that. I'm asking since the answers there still do not allow me to work out an answer to my question, which is a bit more specific. ...
5
votes
0answers
96 views

Spectral Sequences of Parametrized Spectra

I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up: Suppose that I have a parametrized spectra ...
0
votes
0answers
89 views

permutation action on cohomology of configuration space

Let $F(M,n)$ be the $n$-th configuration (ordered) of manifold $M$. In the paper The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres, ...
1
vote
1answer
170 views

permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds. In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...
4
votes
0answers
117 views

When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?

Let $\mathcal V$ be a closed symmetric monoidal model category. It is well known that the category $\mathcal V$-$\mathbf{Cat}$ of $\mathcal V$-enriched categories is itself a closed symmetric monoidal ...
6
votes
2answers
324 views

contractible configuration spaces

Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$. My question: is $F(S^\infty,k)$ ...
0
votes
1answer
211 views

On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
4
votes
0answers
112 views

When does a map in the stable homotopy group gets killed when smashed with cone of itself?

Consider an element $e \in \pi_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence $$ S^n \to S^0 \to C.$$ ...
14
votes
1answer
414 views

Free Loop-Space Recognition Principle

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...
4
votes
1answer
150 views

fibrant generation of $sSet_{Quillen}$?

I wonder if someone has proved that $sSet_{Quillen}$ is not a fibrantly generated model structure ? Do we know something about the possible fibrant generation of $sSet_{Quillen}$ ? Thanks
1
vote
1answer
67 views

Cartesian products between cofibrant simplicial presheaves

Let $Psh(\mathcal{C})$ be the category of simplicial presheaves equipped with the projective model structure. The cartesian product between two representables presheaves is clearly again representable ...
2
votes
0answers
129 views

A Cartesian model structure (and straightening for) on $n$-trivial simplicial sets

A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a ...
17
votes
1answer
437 views

Super-cobordisms

One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
5
votes
1answer
144 views

homotopy tensor product of functors and bar construction

I'm trying to take familiarity with homotopy theory and I have the following questions. Let $\mathcal{C}$ be a small category, and let $F\: : \:\mathcal{C}\to \mathcal{M}$ that take values in a ...