**26**

votes

**2**answers

852 views

### Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...

**2**

votes

**1**answer

197 views

### When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure.
As is ...

**3**

votes

**1**answer

137 views

### homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...

**1**

vote

**0**answers

153 views

### Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence?

Let $i: X \to \mathrm{N}\mathcal C$ be a monomorphism in the category of simplicial sets, with $\mathcal C$ a category and $\mathrm{N}\mathcal C$ its nerve. I am looking for sufficient conditions (and ...

**2**

votes

**1**answer

128 views

### recognising weak equivalences of simplicial sets

$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper Weak Equivalences ...

**1**

vote

**0**answers

103 views

### Properties of “incomplete finite simplicial complexes”

Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...

**6**

votes

**0**answers

138 views

### Connection between quasifibrations and homotopy cartesian squares

Let me first fix the definitions.
A map $p\colon E\rightarrow B$ is called a quasi-fibration, iff the canonical inclusion $p^{-1}(b)\rightarrow hofib_b(p)$ is a weak equivalence for all for all $b\in ...

**9**

votes

**0**answers

261 views

### Homological algebra is linearized homotopical algebra?

I have stumbled across statements like
Homological algebra is linearized homotopical algebra.
Chain complexes are linearizations of simplicial complexes.
The Dold-Kan correspondence was ...

**5**

votes

**0**answers

141 views

### G-spaces and SG-module spectra

This question is related to the one here, but has a slightly different angle.
Let $G$ be a topological group and let $X$ be a $G$-space. Taking the suspension spectrum $\Sigma^{\infty}_+ X$ (in my ...

**0**

votes

**0**answers

66 views

### cohomology of labelled configuration space & relation with braid space

Let $M$ be a manifold and $(X,*)$ be a pointed topological space. ( If we want, we can let $M=S^2,S^1\times \mathbb{R},etc.$)
Let $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$.
...

**7**

votes

**3**answers

355 views

### Mathematical value of constructing sphere eversions

I am extremely impressed by the work that has been done constructing sphere eversions, and other similar explicit geometrical proofs. In particular, surely nobody can fail to be impressed by the ...

**3**

votes

**2**answers

218 views

### cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for ...

**13**

votes

**1**answer

223 views

### Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...

**1**

vote

**1**answer

143 views

### fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$.
...

**7**

votes

**1**answer

349 views

### Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...

**1**

vote

**2**answers

101 views

### Reference for the proof of a neighbourhood characterisation of cofibrations

I am interested in a reference for the proof of the following
theorem for $A,X$ being CGWH topological spaces.
Let $A\subset X$ be a closed subspace,
such that there exists a continuous $\phi : ...

**4**

votes

**1**answer

231 views

### Relation between cohomology of ordered and unordered configuration spaces

Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group ...

**0**

votes

**0**answers

124 views

### Quillen adjunction betwen simplicial presheaves and cochain complexes

Let $sPsh(\mathcal{C})$ the category of simplicial presheaves over a small category $\mathcal{C}$. Let $Ch^{*}_{\geq 0}$ be the category of positively graded cochain complexes of modules over a field ...

**-1**

votes

**1**answer

88 views

### cohomology algebra of unordered configuration space on Euclidean space

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra ...

**4**

votes

**0**answers

131 views

### cohomology algebra of unordered configuration space with coefficients the finite fields

in the paper The cohomology algebra of unordered configuration spaces (Y. Félix, D. Tanré, J. London Math. Soc., 2005), Theorem 4:
Let $M$ be an odd-dimensional, compact, closed, oriented manifold. ...

**1**

vote

**1**answer

111 views

### Does the nerve functor preserve fibrations?

As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ?
By fibration of groupoids I mean a fibration for the ...

**28**

votes

**2**answers

964 views

### Is there a “simplification” functor in algebraic topology?

Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a ...

**3**

votes

**2**answers

270 views

### Can homotopy pullbacks of spaces be checked on fibers?

As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak ...

**7**

votes

**2**answers

214 views

### Ring structure on the $K(1)$-local homotopy of $S^0$ at the prime 2

Let's write $S$ for the $K(1)$-local sphere at the prime 2. Then there is a cofibre sequence
$$S \to KO \to KO$$
where I'm using $KO$ to denote the $K(1)$-localization of orthogonal K-theory, and ...

**5**

votes

**1**answer

207 views

### Mayer-Vietoris sequence for twisted R-homology

In this paper Ando, Blumberg, Gepner, Hopkins and Rezk define the twisted $R$-Homology of a ring spectrum $R$ together with a map $f \colon X \to R$-$Line$ to be
$$
R^f_n(X) =
\pi_0(map_R(\Sigma^nR, ...

**4**

votes

**1**answer

83 views

### Singularity-free isotopies between string diagrams for monoidal categories

Given a monoidal category, it is a theorem of Joyal and Street that an equation between string diagrams is provable from the axioms if and only if there is a recumbent isotopy that relates them. The ...

**9**

votes

**1**answer

274 views

### Example of a saturated class of morphisms which is not _obviously_ saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists ...

**2**

votes

**1**answer

142 views

### Is the discrete nerve of a small category a complete Segal space?

While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly ...

**6**

votes

**1**answer

417 views

### homotopy fixed points and fixed points

Let $X$ a smooth projective scheme over a field $k$. And let $THH(X)$ denotes the topological Hochschild homology of $X$. Recall that the spectra $THH(X)$ admits an action of the of circle $S^{1}$. ...

**0**

votes

**1**answer

188 views

### equivalence in simplicial category

Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...

**23**

votes

**2**answers

726 views

### Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...

**5**

votes

**0**answers

88 views

### Contractibility of a poset-indexed colimit

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ ...

**4**

votes

**1**answer

191 views

### To what extent are homotopy colimits over a weakly contractible category “determined by local data”?

[I'd be very happy for a better question title, if anyone has any suggestions.]
I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a ...

**4**

votes

**2**answers

277 views

### Can one make a category concrete by “enlarging the universe”?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...

**3**

votes

**1**answer

219 views

### Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper:
Cohen called $A^R_n$ "a standard tool used in ...

**6**

votes

**2**answers

299 views

### Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure.
Have the obstructions for an object ...

**10**

votes

**1**answer

214 views

### String Orientation and Level Structures

Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra
$$MSpin \to KO$$
and
$$MSpin^c \to KU.$$
Likewise, but more ...

**0**

votes

**0**answers

55 views

### Lusternik-Schnirelmann category of rational spaces of infinite type

Suppose I have an $n$-dimensional (simply-connected) rational space $X$ with L-S category $k$. Must there be a finite rational subcomplex $K \subseteq X$ with $\mathrm{cat}(K) = k$?
Even better, I ...

**4**

votes

**2**answers

380 views

### $E_n$-space and n-connected pointed space

Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when ...

**6**

votes

**0**answers

110 views

### When is the diagonal inclusion a $\Sigma_2$-cofibration?

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this ...

**7**

votes

**1**answer

307 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

238 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

111 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

226 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

292 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

381 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

275 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

163 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

49 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

648 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 ...