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### 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 ...

**25**

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696 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 ...

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555 views

### Finite complexes whose homotopy groups are not “finitely generated”

I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that ...

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869 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 ...

**20**

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545 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**

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363 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 ...

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306 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 ...

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346 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). ...

**12**

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211 views

### Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...

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127 views

### Besides F_q, for which rings R is K_i(R) completely known?

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...

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443 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 ...

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603 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 $\pi_{19}(SO)$, ...

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547 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 ...

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244 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 ...

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288 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 ...

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359 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 ...

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212 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**

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380 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 ...

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235 views

### From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...

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208 views

### Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...

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135 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 ...

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229 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 ...

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407 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 ...

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296 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 ...

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255 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 ...

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620 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 ...

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433 views

### Homotopy type of complex algebraic varieties

In his 1974 ICM adress "Poids dans la cohomologie des variétés algébriques", Pierre Deligne explains that any finite polyhedron has the same homotopy type as a complex algebraic variety (section 6.).
...

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118 views

### $v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...

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229 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 ...

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152 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?
...

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586 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 ...

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493 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, ...

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312 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 ...

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258 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 ...

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199 views

### Topological quotient Hopf-algebras and “change-of-rings”

One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. ...

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122 views

### Picard-Brauer exact sequence for infinity categories

This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...

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187 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 ...

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220 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 ...

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178 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 $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ ...

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306 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?

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269 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 ...

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222 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 ...

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235 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 ...

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582 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 ...

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260 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 ...

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292 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 ...

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388 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 ...

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216 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 ...

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153 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, ...

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179 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 ...