The homotopy-theory tag has no wiki summary.
8
votes
2answers
458 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
184 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
144 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 ...
3
votes
1answer
377 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
202 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
90 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
139 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
59 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 :
...
8
votes
0answers
447 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 ...
12
votes
1answer
413 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
207 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
200 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
179 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
298 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
59 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
878 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
145 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
188 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
130 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
247 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
586 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
126 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
176 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
99 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
259 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 ...
2
votes
0answers
110 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
101 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
215 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
169 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
237 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
402 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
248 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
142 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
92 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 ...
3
votes
1answer
116 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
152 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
412 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 ...
11
votes
4answers
793 views
Is there a good way to understand the free loop space of a sphere?
I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I ...
1
vote
0answers
100 views
Does compactly supported cohomology make sense for cosimplicial spaces?
As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...
4
votes
1answer
336 views
Homotopy of quivers
The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...
3
votes
1answer
143 views
About (co)limits of accessible categories
I am reading the paper colimits of accessible categories. In the introduction, the authors summarize what is known about limits and colimits of accessible categories. I believed that there was ...
7
votes
0answers
132 views
Notes by Bousfield
I am looking for a copy of "Operations on derived functors of non-additive functors" by Bousfield. It is referenced in many papers and is supposedly from 1967.
Obviously, and electronic copy would be ...
9
votes
1answer
257 views
Exponentiation in finite simplicial sets
A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set ...
11
votes
2answers
512 views
Is every connected space equivalent to some B(Aut(X))?
Given a connected space $B$, is there always some space $X$ with $B \simeq \mathbf{B}(\mathrm{Aut}(X))$?
Here by space I mean simplicial set, by $\mathrm{Aut}(X)$ I mean the simplicial monoid of ...
18
votes
1answer
812 views
Why not a Roadmap for Homotopy Theory and Spectra?
MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
2
votes
0answers
203 views
chain complexes and homotopy limits
Let $\mathbf{Ch}$ be the category of unbounded chain complexes (with the standard model structure ). The differential increases the degree $d_{n}:M_{n}\rightarrow M_{n+1}$. Consider the full ...
5
votes
1answer
119 views
On the obstruction of a sequence of simplicial spaces that is levelwise a fibration
Given a sequence of simplicial spaces (actually bisimplicial sets)
$$F\to E\to B$$
that is level-wise a fibration, then the geometric realisation does not necessarily have to be a fibration.
If I ...
10
votes
1answer
320 views
Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types
I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might ...
