The homotopy-theory tag has no wiki summary.
7
votes
0answers
193 views
I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?
I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...
7
votes
3answers
793 views
Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?
Is there a (non-Abelian) homology theory that realizes the following:
Let $X,Y$ be manifolds with complexes $C(X),C(Y)$.
Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...
22
votes
3answers
1k views
Are the higher homotopy groups of the Hawaiian earring trivial?
The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...
11
votes
0answers
102 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" ...
3
votes
1answer
219 views
sphere bundles over spheres
Localized at an odd prime there is a space $B_k$ which sits in a fibration $S^{2k+2p-3}\rightarrow B_k \rightarrow S^{2k-1}$ and has homology $H_{\ast}(B_k;\mathbb{Z}/p\mathbb{Z})\cong ...
1
vote
1answer
201 views
Possible homotopy-theoretical approach to Gauss-Bonnet
Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
2
votes
2answers
190 views
Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra
Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum ...
6
votes
1answer
189 views
Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?
Consider the category C of pointed simplicial sets.
The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C
models the suspension and loop functors on the underlying ∞-category of C.
There is another ...
15
votes
1answer
393 views
Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?
Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
8
votes
1answer
299 views
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
3
votes
0answers
126 views
Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?
Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
9
votes
2answers
506 views
Vector bundles on vector bundles
Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
7
votes
1answer
199 views
Homotopy of localisations of colimits
Let $X_k$ be a family of spectra equipped with maps $f_k: X_k \to X_{k+1}$. If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy ...
1
vote
1answer
163 views
Is a 'join' of two cofibrations a cofibration?
I have encountered with following problem while I was learning homotopy theory.
Let $A\rightarrow X$ and $B\rightarrow Y$ are cofibrations (in a good category). Is it true that a map $A\times ...
4
votes
1answer
417 views
Topological fundamental group of spec(R)
Let $R$ be a commutative ring with identity. Assume that $X = Spec(R)$ with the Zariski topology.
When is this space path connected? And also we want to know the topological fundamental group of ...
0
votes
1answer
217 views
Computing the fundamental group of a flag variety
Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
2
votes
0answers
104 views
Homotopy limits of weak diagrams
This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor
and to the question here: A homotopy commutative diagram that cannot be ...
0
votes
1answer
151 views
Homotopy limit of a cosimplicial category
Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
...
0
votes
0answers
62 views
Problem on infinite dimensional metric space, with rigidity assumption
By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
...
9
votes
0answers
488 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 ...
13
votes
1answer
449 views
Adams' theorems on the Hopf-Whitehead J-homomorphism
The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$,
a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...
20
votes
4answers
1k views
origin of spectral sequences in algebraic topology
I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).
I have been "brought up" as an algebraic ...
7
votes
2answers
241 views
Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...
1
vote
1answer
213 views
What's the relationship between B Aut(G) and B Aut(BG) for a (discrete) group G?
(Some of the notational choices I"m about to make might be iffy; I'm happy to take suggestions for improvements.)
Let $G$ be a (discrete) group. Think of it as an object in the $2$-category of small ...
9
votes
1answer
200 views
Equivariant homotopy, simplicially
It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...
9
votes
1answer
310 views
Stable moduli interpretation of $\mathbb{R}\mathrm{P}^\infty_{-1}$
I attended a talk recently which closed with the following tantalizing facts: there is a naturally occurring map of spectra $$K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1},$$ which can be ...
1
vote
0answers
61 views
What is a homotopy between bisimplicial maps
I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here.
I am looking for the naive notion of homotopy.
For maps $f, g: A\to B$ of simplicial sets ...
34
votes
2answers
899 views
Is there an analog of Sperner's lemma for the Hopf invariant?
Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic."
My question is, does there exist ...
3
votes
1answer
149 views
Higher homotopy groups of a Zariski closed subset of $\mathbb C^n$
Suppose $V\subset \mathbb C^n$ is a Zariski closed subset. Is it true that the higher homotopy groups $\pi_i(\mathbb C^n-V)$ vanish for $0<i<$ some number depending on the codimension of $V$ in ...
3
votes
0answers
198 views
Homotopy category of groupoids
The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description:
iso-classes of functors.
formally invert equivalence functors (i.e. ...
4
votes
1answer
135 views
Sullivan's $H$-space equivalence between $G/PL[1/2]$ and $BO[1/2]$
There is a theorem by Sullivan of the following form:
Theorem: There is an equivalence of $H$-spaces
$$ G/PL[\tfrac{1}{2}] \simeq BO_{\otimes}[ \tfrac{1}{2} ]\ . $$
It can be found for example ...
6
votes
3answers
288 views
classifying $\infty$-toposes for topological/localic groups?
Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...
8
votes
1answer
623 views
What are simplicial topological spaces intuitively?
(This is a repost of a question from MSE. I hope there is more to say.)
I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial ...
0
votes
0answers
131 views
Thomason’s homotopy colimit theorem for pseudo-functor
Thomason’s homotopy colimit theorem
R W Thomason, Homotopy colimits in the category of small categories,
Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109
says that for a functor $F:I^{op}\to ...
1
vote
1answer
186 views
Massey product in Dual Steenrod Algebra
Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ ...
22
votes
2answers
1k views
Why should I care about topological modular forms?
There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
0
votes
0answers
101 views
What is the homotopy type of $\Sigma \Omega X$?
Let $X$ be a topological space.
We may be able to 1-connected.
Is there general theory about $\Sigma \Omega X$?
I know that James's results for sphere $\Sigma \Omega S^{k+1} \simeq \bigvee ...
2
votes
1answer
279 views
Homotopy groups of filtered homotopy limits
Let $X$ be the homotopy limit of a filtered system of simplicial sets $X_i$. When are the morphisms $\pi_j(X)\to \varprojlim \pi_j(X_i)$ surjective for all $j\ge 0$? This seems to be no problem when ...
3
votes
0answers
114 views
What are the Čech-local equivalences of (simplicial pre)sheaves?
Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...
1
vote
0answers
104 views
Model structure on stacks
does anyone know if there is a model structure on stacks where the cofibrations are the monomorphisms ?
As far as I know usually to get a model structure on stacks one localizes a model structure on ...
2
votes
1answer
220 views
Homotopy versus path-homotopy on punctured surface
I have some problems with homotopies.
The situation is this:
Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
3
votes
0answers
173 views
What is a Homotopy between $L_\infty$-algebra morphisms II
I would like to continue on question I asking, what is a homotopy between
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ...
3
votes
1answer
255 views
What is the “higher version” of chain homotopy in singular homology?
In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
11
votes
0answers
421 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 ...
6
votes
2answers
259 views
Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)
The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak ...
5
votes
0answers
146 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, ...
6
votes
0answers
98 views
Can any suspension spectrum be realized as Waldhausen K-theory?
If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the ...
4
votes
1answer
129 views
Is the injective structure on unbounded chain complexes simplicial?
In Mark Hovey's article Model category structures on chain complexes of sheaves (arXiv:math/9909024) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian ...
3
votes
1answer
155 views
Factorization of morphisms in a diagram category
Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category ...
15
votes
2answers
438 views
Distinct manifolds with the same configuration spaces?
For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$.
What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...
