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33
votes
0answers
728 views

Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...
26
votes
0answers
675 views

Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting) I start by recalling the analytic definition of KO-theory: The following ...
21
votes
0answers
846 views

On the (derived) dual to the James construction.

Background If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by $$ X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots $$ in ...
19
votes
0answers
483 views

Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?

For a (discrete) monoid $M$, the classifying space $BM$ is the geometric realization of the nerve of the one object category whose hom-set is $M$. (This definition gives the usual classfiying space ...
15
votes
0answers
350 views

Algebraic closure as a fibrant replacement?

Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...
15
votes
0answers
295 views

Lipschitz constant of a homotopy

Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole. A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family off maps $h_x\colon M\to ...
15
votes
0answers
341 views

Other examples of computations using transfer of structure from the chains to the homology?

There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). ...
11
votes
0answers
419 views

About maps inducing bijections on homotopy classes

Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
11
votes
0answers
598 views

What's so difficult about $\pi_{15}(SO)$?

Regarding the table of $SO(n)$s-of-origin in Davis+Mahowald (if you can get MathSciNet), is there a good reason that it should take longer for $\pi_{15}(SO)$ to be representable than $\pi_{19}(SO)$, ...
10
votes
0answers
284 views

Hilton-Eckmann dual of the Steenrod Algebra

In essence my question can be stated as follows: fill in the analogy $$ \text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product} $$ $$ \updownarrow \qquad\qquad ...
10
votes
0answers
354 views

Mapping cylinders of fibrations

If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder of $p$ also a fibration? I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The ...
9
votes
0answers
485 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...
9
votes
0answers
200 views

The spectral sequence of a tower of principal fibrations

Assume we have a tower of fibrations (of simplicial sets, let's say): $$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$ Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
9
votes
0answers
376 views

Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$: 1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...
8
votes
0answers
230 views

When is the one-point compactification well-pointed?

This is a follow up to my previous question. Question: Is there a reasonably natural set of conditions which guarantee that the one-point compactification $X^+$ of a locally compact Hausdorff ...
8
votes
0answers
215 views

Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem

$\newcommand\Ext{\mathrm{Ext}} \newcommand\Z{\mathbb{Z}} \newcommand\G{\mathbb{G}}$ The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the ...
8
votes
0answers
389 views

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title. In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...
8
votes
0answers
254 views

Question about $A_\infty$ maps

Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping ...
8
votes
0answers
617 views

Is a functor which is a sheaf for open covers and finite closed covers automatically a sheaf for covers by simplices?

Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for ...
7
votes
0answers
133 views

Notes by Bousfield

I am looking for a copy of "Operations on derived functors of non-additive functors" by Bousfield. It is referenced in many papers and is supposedly from 1967. Obviously, and electronic copy would be ...
7
votes
0answers
220 views

The spheres operad

I have a rather naive question. Consider the space of all maps $$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$ for all possible natural numbers $n, k, j_1, \cdots , j_k$. This ...
7
votes
0answers
142 views

Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting? ...
7
votes
0answers
526 views

Homotopy groups of the orthogonal group

I'm interested in knowing what $n$-dimensional vector bundles on the $n$-sphere look like, or equivalently in determining $\pi_{n-1}(SO(n))$; here's a case that I haven't been able to solve. Let $n ...
7
votes
0answers
292 views

A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
7
votes
0answers
483 views

toy examples of equivariant homotopy theory

I've heard a little recently about equivariant homotopy theory, and so I decided to try out some baby examples just to get a feel for it. I'm not even sure if these are the right thing to look at, ...
7
votes
0answers
306 views

Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology. The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
7
votes
0answers
256 views

The residue class functor from a Frobenius category to its stable category induces a functor on cube-shaped diagrams - is it dense?

Let $\mathcal{E}$ be a Frobenius category, i.e. an exact category with sufficiently many bijective objects. (Such as e.g. the category of complexes over an additive category.) Let ...
6
votes
0answers
97 views

Can any suspension spectrum be realized as Waldhausen K-theory?

If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the ...
6
votes
0answers
182 views

What is the state of the art in 4-manfold 2-types?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
6
votes
0answers
169 views

When is the cohomology cross product square nonzero?

Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ ...
6
votes
0answers
302 views

Why the $M$ for Thom spaces?

I've heard $E$ is for entire space, $B$ is for base space, so what is $M$ for?
6
votes
0answers
219 views

Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...
6
votes
0answers
221 views

Completion of n-fold Segal spaces

During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for ...
6
votes
0answers
581 views

An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point: Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with the contravariant model structure (cofibrations are monomorphisms ...
6
votes
0answers
256 views

homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
6
votes
0answers
382 views

The Mapping Cylinder of a Pullback Square

Suppose I have a pullback square, which I think of as a map from the fibration $q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$ from the mapping cylinder $M$ of ...
6
votes
0answers
214 views

Do p-compact groups have a sufficiently good notion of “flag variety” and “intersection cohomology”?

This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think. Background An outstanding problem ...
5
votes
0answers
146 views

Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1. If necessary, the model structures can be assumed to be simplicial, ...
5
votes
0answers
153 views

Homotopy unites of a differential graded algebra

I apologize in advance if the question is too basic. Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An ...
5
votes
0answers
165 views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite ...
5
votes
0answers
123 views

Refined cofinality theorem for homotopy limits of spaces

If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow ...
5
votes
0answers
154 views

Motivic homotopy spectral sequence

I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ ...
5
votes
0answers
206 views

Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$. The ...
5
votes
0answers
198 views

Mod 3 Moore spectrum

I only know through stories that mod 3 moore spectrum is not associative. I do not know of any proof. I have been informed that Toda had proved it in the paper "Extended $p^{th}$ power". I was not ...
5
votes
0answers
161 views

Reference request: splittings in rational homotopy theory

It is well known that for simply-connected rational spaces, every suspension splits as a wedge of rational spheres and every loop space splits as a product of rational Eilenberg-Mac Lane spaces. ...
5
votes
0answers
255 views

Generating acyclic cofibrations for the various model structures in higher category theory

There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example: The Joyal model structure on ...
5
votes
0answers
126 views

Joins and classifying spaces in the category of compactly generated spaces

In Milnor's Construction of Universal Bundles, II, he defines $E_nG$ by repeated joins of $G$ with itself, but he has to use the `strong topology' on the join instead of the everyday topology that ...
5
votes
0answers
289 views

The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes. Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
5
votes
0answers
296 views

How does the cokernel of the J-homomorphism count exotic spheres?

The wikipedia article on the J-homomorphism says that "the cokernel of the J-homomorphism is of interest for counting exotic spheres". I'd like to think this makes some sort of philosophical sense; ...
5
votes
0answers
318 views

Why are Delta-generated spaces locally presentable?

Does anybody understand why Delta-generated spaces are locally presentable? This is of course claimed by Jeff Smith, and there is a paper by Fajstrup and Rosicky A convenient category for directed ...