**7**

votes

**0**answers

93 views

### Free generators for the fat commutator subgroup

There is a homomorphism $\langle x,y\rangle\to\langle x\rangle$ of free groups, sending $y$ to $1$. We can combine this with the other obvious homomorphism to get a surjective homomorphism
$$ ...

**8**

votes

**1**answer

200 views

### Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?

Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...

**3**

votes

**1**answer

167 views

### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this.
I wonder if there is any source telling about a ...

**10**

votes

**2**answers

532 views

### Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the ...

**7**

votes

**1**answer

232 views

+200

### Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...

**3**

votes

**1**answer

121 views

### very basic question on p-typical group law and MU and BP

I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU_{(p)} \to MU_{(p)}$ described in his classic paper 5-page paper On formal group laws...
Let $F$ be a formal group law over a ...

**5**

votes

**1**answer

317 views

### Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...

**1**

vote

**0**answers

77 views

### Sufficient criteria for a nerve of a topological category to be good

I know that the following statement is true and I am looking for a reference:
Given a topological category $\mathcal C$ (i.e. morphisms and objects form a space and all maps in the definition of a ...

**7**

votes

**1**answer

171 views

### Are $E_n$-operads not formal in characteristic not equal to zero?

This is a short question:
Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?

**3**

votes

**2**answers

264 views

### Is it possible to compute coefficients of the formal group of an elliptic curve?

This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the ...

**5**

votes

**0**answers

138 views

### On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...

**17**

votes

**2**answers

468 views

### Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...

**6**

votes

**0**answers

177 views

### How do the direct and inverse image sheaf functors interact with homotopy?

This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by ...

**13**

votes

**1**answer

184 views

### Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphisms of the first

In a previous Mathoverflow question, we saw that the fundamental group of the space $Imm(S^2,\mathbb{R}^3)$ of immersions the 2-sphere in ordinary 3-space is isomorphic to $\mathbb{Z}/2 \times ...

**12**

votes

**1**answer

292 views

### Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...

**1**

vote

**0**answers

55 views

### Geometric Realizations of Simplicial Based Spaces

Suppose I have two simplicial based topological spaces $X_\bullet$ and $Y_\bullet$, and the degeneracy maps of each satisfy the based homotopy extension property (but not necessarily the unbased ...

**7**

votes

**1**answer

316 views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**7**

votes

**0**answers

104 views

### How stable is the top cell of Lie group?

It is well know fundamental class of a compact lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence ...

**7**

votes

**1**answer

187 views

### Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...

**4**

votes

**4**answers

458 views

### When is the quotient by an $n$-fold loop space an $m$-fold loop space?

Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...

**26**

votes

**1**answer

514 views

### Classifiying sphere eversions

For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as ...

**1**

vote

**0**answers

51 views

### cohomology ring of iterated loop-suspension of spheres

In the book The unstable Adams spectral sequence for free iterated loop spaces (http://www.ams.org/bookstore?fn=20&arg1=memoseries&ikey=MEMO-36-258), Corollary 3.14:
Question: given $n$, ...

**2**

votes

**1**answer

89 views

### dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32
Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...

**18**

votes

**1**answer

331 views

### Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...

**4**

votes

**1**answer

121 views

### cohomology ring of infinite iterated loop space

What is the cohomology ring
$$
H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)?
$$
I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...

**2**

votes

**0**answers

68 views

### cohomology ring of base-point-preserving maps on the 3-sphere

I find that $\text{Map}_*(S^3;S^3)=\Omega^3S^3$. I want to find the cohomology ring of $H^*(\Omega^3S^3;\mathbb{Z}_2)$.
In the paper On configuration spaces, their homology, and Lie groups, I find ...

**4**

votes

**1**answer

91 views

### unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then
$$
B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1},
$$
$$
B(S^n,2)\simeq \mathbb{R}P^n.
$$
Hence
$
(*)
$
$$
...

**4**

votes

**1**answer

173 views

### Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...

**26**

votes

**2**answers

1k views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...

**39**

votes

**1**answer

1k views

### Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...

**0**

votes

**1**answer

208 views

### p-local space vs p-completion.

I have some trouble to understand the difference between the p-completion and p-local space.
if $X$ a simply connected spaces such that all higher homotopy groups are finitely generated groups then ...

**5**

votes

**1**answer

187 views

### Representing classes in *relative* homology by submanifolds

There are nice results for representing homology classes by submanifolds, in particular for any class in $H_i(X)$ with $i\le 6$, see here. When $X$ is low-dimensional I can start getting explicit, but ...

**2**

votes

**1**answer

83 views

### Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors
$F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...

**3**

votes

**1**answer

137 views

### Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...

**3**

votes

**1**answer

211 views

### Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits.
Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...

**11**

votes

**2**answers

231 views

### Quasicategories for non-simplicial model categories

If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...

**3**

votes

**1**answer

227 views

### coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...

**5**

votes

**1**answer

471 views

### reference for “Topological algebra of Grothendieck”

I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra,
with special emphasis on topoi.

**2**

votes

**0**answers

116 views

### the homology of configuration spaces [closed]

In the paper ON THE HOMOLOGY OF CONFIGURATION SPACES, Section 5.4,
Why $C(M\times \mathbb{R};X)\simeq \Omega C(M; SX)$? Does this hold for $X$ not connected?

**8**

votes

**0**answers

313 views

### Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...

**3**

votes

**1**answer

144 views

### Is a pullback along a Dold fibration a homotopy pullback?

Let $$
\begin{array}{ccc}
A & \to & B
\cr\downarrow&&\downarrow
\cr
A'& \to &B'
\end{array}
$$ be a pullback square in the category of all topological spaces (not just in a ...

**1**

vote

**1**answer

109 views

### Does a topological hypercover always have free degeneracies?

This question arises when I am reading Dugger and Isaksen's "Hypercovers in topology". According to Definition 4.1 in that paper, A hypercover of a space $X$ is an augmented simplicial space $U_*\to ...

**0**

votes

**0**answers

97 views

### reference for groupoid cohomology

In nLab (groupoid cohomology) says:
"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."
Are there references for ...

**19**

votes

**1**answer

553 views

### Is there a generalization of homotopy groups to fractional dimensions

Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?

**7**

votes

**1**answer

396 views

### Loop space generalization

Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$
is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space ...

**3**

votes

**0**answers

233 views

### Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...

**0**

votes

**1**answer

311 views

### Dimension of two homotopy equivalent manifolds [closed]

Let $M,N$ be a closed (connected, without boundary, say smooth) manifolds which are homotopy equivalent. Does it follows that they are of the same dimension? One should be aware of examples of ...

**1**

vote

**0**answers

384 views

### Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?

**2**

votes

**0**answers

63 views

### section spaces related to configuration spaces

In the paper Configuration spaces of positive and negative particles, Dusa McDuff, a section space $\Gamma(M)$ is constructed:
And in the paper ON THE HOMOLOGY OF CONFIGURATION SPACES. C.-F. ...

**2**

votes

**0**answers

99 views

### homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$
$$
...