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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218 views

Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?

This question has been inspired by an answer to the question Reference request: Topology on the space of smooth compact submanifolds; I've asked it in a comment to that answer but then decided to make ...
2
votes
1answer
154 views

Two-bridge knots and CW-complex

The fundamental group of any two-bridge knot K in $\mathbb{S}^3$ has a presentation with two generators and one relation. On the other hand, it's possible to provide a CW-complex with only one 0-cell ...
28
votes
2answers
1k views

Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...
2
votes
1answer
225 views

When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure. As is ...
3
votes
1answer
150 views

homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold. Let $F(M,n)$ be the configuration space of $n$-tuples on $M$. Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...
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0answers
161 views

Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence?

Let $i: X \to \mathrm{N}\mathcal C$ be a monomorphism in the category of simplicial sets, with $\mathcal C$ a category and $\mathrm{N}\mathcal C$ its nerve. I am looking for sufficient conditions (and ...
2
votes
1answer
137 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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0answers
106 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
151 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 ...
11
votes
0answers
326 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
votes
0answers
150 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 ...
1
vote
0answers
86 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
votes
3answers
376 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
240 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 ...
14
votes
1answer
257 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
174 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 $G$. ...
7
votes
1answer
371 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
vote
2answers
107 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 : ...
5
votes
1answer
261 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
144 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 ...
-1
votes
1answer
107 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 ...
5
votes
0answers
147 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
vote
1answer
129 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 ...
28
votes
2answers
1k 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 ...
4
votes
2answers
313 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
235 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
230 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
90 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
287 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
153 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
427 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
200 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
779 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
95 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
205 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
299 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
221 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
357 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
236 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 ...
0
votes
0answers
62 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
421 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 ...
7
votes
0answers
120 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
324 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
263 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
113 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
235 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? ...
7
votes
1answer
306 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
396 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 ...
2
votes
2answers
349 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 ...
8
votes
0answers
177 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 ...