**7**

votes

**1**answer

309 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

**1**answer

240 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

**1**answer

112 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

**1**answer

229 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

**1**answer

295 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

**0**answers

384 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

**2**answers

276 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 ...

**7**

votes

**0**answers

166 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 ...

**2**

votes

**0**answers

52 views

### Cell(J) vs Cof(J) in $\text{sSet}_{\text{Quillen}}$

consider sSet equipped with its Quillen model structure $\text{sSet}_{\text{Quillen}}$, we know that a trivial cofibration is a retract of a transfinite composition of pushouts of horn inclusions. I ...

**15**

votes

**2**answers

657 views

### When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...

**1**

vote

**0**answers

94 views

### Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...

**2**

votes

**1**answer

194 views

### Simplicial version of the A-infinity operad

I am looking for a description of the $A_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial ...

**2**

votes

**1**answer

343 views

### fixed point and homotopy fixed points

Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed).
Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$,
If I'm not wrong Miller-Lannes,... theory provides tools and ...

**10**

votes

**1**answer

464 views

### Is there a category whose isomorphisms are precisely the simple homotopy equivalences?

If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...

**11**

votes

**2**answers

681 views

### from a circle to higher spheres

Question: Is there a group $G$ and a CW-complex $X$ such that
1) $X$ is homotopy equivalent to the circle $S^{1}$.
2) $G$ acts on $X$
3) the space of fixed points $X^{G}$ is weakly equivalent to ...

**5**

votes

**1**answer

284 views

### Framed version of braided monoidal category

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...

**6**

votes

**2**answers

192 views

### Are all unstable homotopy groups of $U(n)$ torsion?

The first few unstable homotopy groups of the unitary groups $U(n)$ were calculated by Borel-Hirzebruch, Toda, and Kervaire, and they are all torsion. There is a paper by Matsunaga (details below) in ...

**0**

votes

**1**answer

114 views

### Unseparability of two linked rings in higher dimensions [closed]

I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked".
I wonder that is there any similar results for two copies of ...

**12**

votes

**2**answers

627 views

### Eilenberg-Mac lane spaces and a generalization

Let $G$ and $H$ be two abelian groups and let $n>1, m>1$ be two different integers. How many different spaces $X$ (up to homotopy) do we have with the property $\pi_{n} X=G$ , $\pi_{m} X=H$ and ...

**4**

votes

**1**answer

192 views

### Does “simplicial” commute with “Bousfield localization”?

Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect ...

**3**

votes

**1**answer

363 views

### $E_{\infty}$ spaces are $A_{\infty}$ spaces

While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...

**8**

votes

**1**answer

166 views

### Freely adding degeneracies does not change the homotopy type

Background: Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already ...

**11**

votes

**1**answer

286 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 ...

**30**

votes

**4**answers

2k 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 ...

**7**

votes

**1**answer

228 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: ...

**15**

votes

**0**answers

270 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

**1**answer

219 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 ...

**8**

votes

**0**answers

102 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 ...

**21**

votes

**4**answers

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 ...

**4**

votes

**1**answer

295 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

**0**answers

171 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.

**15**

votes

**2**answers

787 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

**0**answers

106 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

**0**answers

108 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

**1**answer

92 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

**0**answers

155 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

**0**answers

264 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 ...

**5**

votes

**0**answers

166 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

**0**answers

80 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

**0**answers

115 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

**1**answer

138 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

**1**answer

73 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

**1**answer

228 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 ...

**12**

votes

**2**answers

477 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 ...

**4**

votes

**0**answers

142 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 ...

**4**

votes

**1**answer

215 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" ...

**2**

votes

**0**answers

75 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

**0**answers

82 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

**0**answers

67 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

**1**answer

159 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 ...