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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92 views

recognising weak equivalences of simplicial sets

$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper Weak Equivalences ...
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91 views

Properties of “incomplete finite simplicial complexes”

Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K. ...
6
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0answers
121 views

Connection between quasifibrations and homotopy cartesian squares

Let me first fix the definitions. A map $p\colon E\rightarrow B$ is called a quasi-fibration, iff the canonical inclusion $p^{-1}(b)\rightarrow hofib_b(p)$ is a weak equivalence for all for all $b\in ...
8
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0answers
243 views

Homological algebra is linearized homotopical algebra?

I have stumbled across statements like Homological algebra is linearized homotopical algebra. Chain complexes are linearizations of simplicial complexes. The Dold-Kan correspondence was ...
5
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0answers
136 views

G-spaces and SG-module spectra

This question is related to the one here, but has a slightly different angle. Let $G$ be a topological group and let $X$ be a $G$-space. Taking the suspension spectrum $\Sigma^{\infty}_+ X$ (in my ...
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54 views

cohomology of labelled configuration space & relation with braid space

Let $M$ be a manifold and $(X,*)$ be a pointed topological space. ( If we want, we can let $M=S^2,S^1\times \mathbb{R},etc.$) Let $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$. ...
7
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3answers
339 views

Mathematical value of constructing sphere eversions

I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the ...
3
votes
2answers
210 views

cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for ...
13
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1answer
196 views

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

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...
1
vote
1answer
129 views

fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$ G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
6
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1answer
289 views

Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...
1
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2answers
94 views

Reference for the proof of a neighbourhood characterisation of cofibrations

I am interested in a reference for the proof of the following theorem for $A,X$ being CGWH topological spaces. Let $A\subset X$ be a closed subspace, such that there exists a continuous $\phi : ...
4
votes
1answer
200 views

Relation between cohomology of ordered and unordered configuration spaces

Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group ...
0
votes
0answers
116 views

Quillen adjunction betwen simplicial presheaves and cochain complexes

Let $sPsh(\mathcal{C})$ the category of simplicial presheaves over a small category $\mathcal{C}$. Let $Ch^{*}_{\geq 0}$ be the category of positively graded cochain complexes of modules over a field ...
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votes
1answer
81 views

cohomology algebra of unordered configuration space on Euclidean space

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents): Line 2: ... is used to compute the precise algebra ...
4
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0answers
118 views

cohomology algebra of unordered configuration space with coefficients the finite fields

in the paper The cohomology algebra of unordered configuration spaces (Y. Félix, D. Tanré, J. London Math. Soc., 2005), Theorem 4: Let $M$ be an odd-dimensional, compact, closed, oriented manifold. ...
1
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1answer
98 views

Does the nerve functor preserve fibrations?

As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ? By fibration of groupoids I mean a fibration for the ...
25
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1answer
730 views

Is there a “simplification” functor in algebraic topology?

Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a ...
2
votes
2answers
219 views

Can homotopy pullbacks of spaces be checked on fibers?

As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak ...
7
votes
2answers
204 views

Ring structure on the $K(1)$-local homotopy of $S^0$ at the prime 2

Let's write $S$ for the $K(1)$-local sphere at the prime 2. Then there is a cofibre sequence $$S \to KO \to KO$$ where I'm using $KO$ to denote the $K(1)$-localization of orthogonal K-theory, and ...
5
votes
1answer
200 views

Mayer-Vietoris sequence for twisted R-homology

In this paper Ando, Blumberg, Gepner, Hopkins and Rezk define the twisted $R$-Homology of a ring spectrum $R$ together with a map $f \colon X \to R$-$Line$ to be $$ R^f_n(X) = \pi_0(map_R(\Sigma^nR, ...
4
votes
1answer
75 views

Singularity-free isotopies between string diagrams for monoidal categories

Given a monoidal category, it is a theorem of Joyal and Street that an equation between string diagrams is provable from the axioms if and only if there is a recumbent isotopy that relates them. The ...
9
votes
1answer
262 views

Example of a saturated class of morphisms which is not _obviously_ saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...
2
votes
1answer
131 views

Is the discrete nerve of a small category a complete Segal space?

While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly ...
6
votes
1answer
402 views

homotopy fixed points and fixed points

Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. ...
0
votes
1answer
179 views

equivalence in simplicial category

Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...
23
votes
2answers
704 views

Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...
5
votes
0answers
85 views

Contractibility of a poset-indexed colimit

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...
4
votes
1answer
183 views

To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.] I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...
4
votes
2answers
266 views

Can one make a category concrete by “enlarging the universe”?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...
3
votes
1answer
217 views

Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper: Cohen called $A^R_n$ "a standard tool used in ...
6
votes
2answers
267 views

Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. Have the obstructions for an object ...
10
votes
1answer
207 views

String Orientation and Level Structures

Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra $$MSpin \to KO$$ and $$MSpin^c \to KU.$$ Likewise, but more ...
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0answers
54 views

Lusternik-Schnirelmann category of rational spaces of infinite type

Suppose I have an $n$-dimensional (simply-connected) rational space $X$ with L-S category $k$. Must there be a finite rational subcomplex $K \subseteq X$ with $\mathrm{cat}(K) = k$? Even better, I ...
4
votes
2answers
355 views

$E_n$-space and n-connected pointed space

Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when ...
6
votes
0answers
103 views

When is the diagonal inclusion a $\Sigma_2$-cofibration?

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this ...
7
votes
1answer
297 views

Computing naive algebraic singular homology

I'm not sure if this counts as research level, since it might just be an expression of my ignorance. But anyway. Let $\Delta^n_A = Spec(A[X_0, \dots, X_n]/\sum X_i - 1)$ denote the standard algebraic ...
4
votes
1answer
218 views

Is the classifying space of a symmetric monoidal category an infinite loop space?

Wikipedia states: The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an infinite loop space. If my mind is correct, Segals delooping machine gives a ...
1
vote
1answer
109 views

Request a paper by Fred Cohen

I am looking for the following paper by Cohen, F. R.: On combinatorial group theory in homotopy. Homotopy theory and its applications (Cocoyoc, 1993), 57–63, Contemp. Math., 188, Amer. Math. ...
13
votes
1answer
215 views

Homogeneous spaces that are homotopy tori

Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group? ...
6
votes
1answer
278 views

Thom isomorphism from the ABGHR perspective

In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is ...
7
votes
0answers
370 views

Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action

Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$. Consider the $\infty$-category ...
3
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2answers
220 views

When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...
7
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0answers
152 views

Whiskering a monad

In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...
2
votes
0answers
47 views

Cell(J) vs Cof(J) in $\text{sSet}_{\text{Quillen}}$

consider sSet equipped with its Quillen model structure $\text{sSet}_{\text{Quillen}}$, we know that a trivial cofibration is a retract of a transfinite composition of pushouts of horn inclusions. I ...
15
votes
2answers
610 views

When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...
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0answers
91 views

Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...
2
votes
1answer
185 views

Simplicial version of the A-infinity operad

I am looking for a description of the $A_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial ...
2
votes
1answer
310 views

fixed point and homotopy fixed points

Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed). Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$, If I'm not wrong Miller-Lannes,... theory provides tools and ...
10
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1answer
432 views

Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...