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10
votes
1answer
133 views

Diffeomorphisms and homotopy equivalences sliced over BO(n)

There are some classical results stating sufficient conditions on a manifold $\Sigma$ such that every homotopy equivalence $\Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$ is homotopic to ...
-2
votes
0answers
101 views

Is the configuration space of infinite sphere contractible? [on hold]

Let $\Sigma$ be suspension. Let $S^0$ be $0$-sphere. Let $\Sigma^\infty S^0$ be the union $\cup_n \Sigma^n S^0$ with respect to the inclusion $\Sigma^kS^0\subset \Sigma^{k+1}S^0$. Let ...
26
votes
4answers
1k views

Why is it so hard to compute $\pi_n(S^n)$?

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...
6
votes
1answer
142 views

cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$. Let $\rho: ...
0
votes
0answers
59 views

Homotopy uniqueness of some coends

I am not very convinced about Hirschhorn's proof of Corollary 18.4.15 Here, $\mathscr{C}$ is a Reedy category and $\mathcal{M}$ is a simplicial model category. Furthermore, $X\otimes_{\mathscr{C}} K$ ...
13
votes
0answers
192 views

Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect. The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
3
votes
1answer
172 views

Motion planning algorithm

Consider a path connected topological space $X$, one can equip its path space $PX=\{ \gamma: [0,1] \longrightarrow X \; continuous\}$ with the compact open topology. We call a motion planning ...
7
votes
0answers
71 views

What suffices to check completeness in an n-fold Segal space?

Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the Segal condition: For each $j$, the map $$ X_j \to ...
19
votes
4answers
2k views

Why the Dold-Thom theorem?

Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$ It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
3
votes
1answer
240 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
0
votes
0answers
147 views

Goodwillie tower of $\Omega^n$?

What are layers of the Goodwillie tower of the functor "n-th iterated loop space" from based spaces to based spaces? I know the answer for n=0.
13
votes
2answers
666 views

Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra? Is it easier if we impose any of the three conditions: characteristic zero; ...
2
votes
0answers
91 views

When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed by deriving the product functor sSetW→sSet, for example, by performing the componentwise fibrant replacement using Kan's ...
5
votes
0answers
90 views

Explicit generators for homotopy groups of Lie groups

I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$. It is ...
2
votes
1answer
90 views

Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research. Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...
8
votes
0answers
127 views

Reference for maps whose pushouts are also homotopy pushouts

Consider a category C with weak equivalences, e.g., a model category. For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
4
votes
0answers
204 views

Commutation of simplicial homotopy colimits and homotopy products in spaces

Edit: The claim below is wrong, as explained in the comments, because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...
4
votes
0answers
146 views

Getting a Postnikov Tower from the Tot-tower?

I have had a problem I've been thinking of recently but can't seem to make anything of it. Let $X$ be a simplicial set. It is well known that one can construct a Postnikov tower $$P_nX \rightarrow ...
1
vote
0answers
58 views

When localizing a category at a multiplicative system, is there a lemma utilizing Ore Condition/Cancellation once for all roof-independence proofs?

For example, given an additive category $\mathcal{C}$ with a (both sided) multiplicative system S, the localization $S^{-1}\mathcal{C}$ is also additive. Yet to prove that the addition of morphisms in ...
5
votes
0answers
103 views

The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities

The Bullet-Macdonald identity (c.f. On the Adem relations)is the following: $$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$ where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the Adem ...
9
votes
1answer
128 views

When is a continous $\epsilon$-isometry of the sphere surjective?

Equip $\mathbb S^n$ with the standard round metric. Let $f : \mathbb S^n \to \mathbb S^n$ be a continous map satisfying $\vert d(f(x),f(y)) - d(x,y)\vert \leq \epsilon$. Is $f$ is surjective for all ...
3
votes
1answer
64 views

Categories of spans from categories of fibrant objects

Let $\mathscr{C}$ be a category of fibrant objects. For objects $X$, $Y$ of $\mathscr{C}$ we consider the category $\underline{\text{Hom}}(X,Y)$ of spans $X\leftarrow X'\rightarrow Y$, where $X'\to ...
9
votes
1answer
196 views

Morava modules and completed $E$-homology

Let $E = E_n$ be the $n$-th Morava $E$-theory and let $\{ M_{I} \}$ be a tower of generalised Moore spectra. Then (see this previous question) there is a Milnor exact sequence $$0 \to \varprojlim_I ...
10
votes
2answers
398 views

“Economic” CW-structure for Eilenberg-MacLane spaces?

The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
3
votes
0answers
123 views

Flat Connections on the Cotangent Complex

I'm trying to find a reference which defines and discusses some properties of connections and flat connections on the cotangent complex in a homotopical setting. That is to say, a connection or flat ...
3
votes
1answer
186 views

$t$-structure on modules over highly structured ring spectra

Let $Sp$ be a suitably nice model of a suitably nice category of spectra. By this I mean that $Sp$ is a symmetric monoidal closed stable model category, where we can make sense of homotopy "groups" ...
1
vote
0answers
57 views

Latching space functor in Reedy model strucutre

I am new to MO and I hope that this question is suitable for it. Following Hovey's "Model Categories" ( http://ericmalm.net/ac/projects/symmetric-spectra/hovey--model-cats.pdf ) to study model ...
3
votes
0answers
75 views

Large co-H-spaces

I'm searching for examples of co-H-spaces that are not suspensions and that do not admit a finite cone decomposition with respect to the collection of finite type wedges of spheres. We have many ...
2
votes
0answers
65 views

Why does $\pi_t$ preserve pullbacks in this special case?

Let $X$ be a fibrant pointed cosimplicial space. Following Bousfield-Kan, let $\text{lim}^{\partial \Delta_{n+1}} X = M^{n}X$ be the nth matching object of $X$. Onecan then show that there is a ...
4
votes
1answer
136 views

In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$. I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...
2
votes
1answer
276 views

The topology of the classifying space of U(n)

In $\textit{Atiyah and Bott: The Yang-Mills equations over riemann surfaces}$, there is a sentence about the topology of BU(n) (i.e. the classifying space of U(n)): Here $K(\mathbb{Z},n)$ means the ...
4
votes
2answers
320 views

Aspheric functors and Grothendieck fibrations

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, ...
3
votes
1answer
328 views

Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?

The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an E∞-operad; algebras in spaces over the Barratt-Eccles operad model E∞-spaces, i.e., homotopy ...
0
votes
0answers
48 views

Homotopy addition theorem and lifting in a certain diagram

If I am confusing somewhere here, please excuse me and ask me to clarify. The proof I am having trouble with is lemma 1.19 of Goerss-Jardine, pg. 398. I have tried to isolate the troublesome ...
3
votes
0answers
208 views

Quotes from Connes

I found the following remark by Connes HERE: "the main quality of the homotopy type of a manifold is to satisfy Poincare duality not only in ordinary homology but in K-homology with the Fredholm ...
1
vote
1answer
266 views

questions on steenrod algebra

I plan to give a talk on Steenrod algebra in a student seminar. but there are some questions that I didn't find an answer to, and it seems to me that I'm missing something: if the algebra of ...
2
votes
1answer
135 views

what is the stabilization of pointed sets?

Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab). The famous example is ...
10
votes
1answer
454 views

Homology theory represented by Madsen-Tillman spectra

The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of ...
4
votes
1answer
220 views

Higher Degree Data in a Cosimplicial Quasicategory and Delooping

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference. My question is regarding accessing data ...
1
vote
0answers
217 views

Higher Homotopy Groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
9
votes
1answer
346 views

Nonunital $E_\infty$-rings

An elementary fact of algebra is that the category of nonunital commutative rings is equivalent to that of $\mathbb{Z}$-augmented unital commutative rings, the equivalence being given by forming ...
8
votes
1answer
258 views

Clutching functions and Classifying maps

Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, ...
2
votes
1answer
85 views

Does the right adjoint of the category of simplices functor is “homotopicaly inverse” to the category of simplices functor?

Short Version (the question) Let $\text{Cat}$ be the category of (small) categories and $\text{sSet}$ the category of simplicial sets. There is a functor $\Gamma:\text{sSet}\to \text{Cat}$ that takes ...
9
votes
0answers
220 views

Other examples of the algebro-geometric Ran space

First off, sorry if this seems vague. Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...
3
votes
2answers
348 views

Zigzags and contractibility of categories

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion: If there is ...
5
votes
1answer
234 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
1
vote
0answers
102 views

Cell attachment in rational homotopy theory

In Rational Homotopy Theory, there is a model of cell-attachment. In the book "Rational Homotopy Theory", the model is given for attaching only one cell, which is: If $X$ is simply connected space ...
2
votes
1answer
210 views

Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $h\circ i:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...
16
votes
2answers
798 views

How much of homotopy theory can be done using only finite topological spaces?

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...
5
votes
0answers
217 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where ...