Questions tagged [homotopy-theory]
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
3,001
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How are these algebraic and geometric notions of homotopy of maps between manifolds related?
Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...
- 53.1k
5
votes
1
answer
572
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Delooping maps between H-spaces
Hi,
this question is related to my question here. Suppose, I have a topological group $G$ and an $A_{\infty}$-space $H$, which is a CW-complex. Furthermore, I have a map $\varphi \colon G \to H$, ...
- 7,537
2
votes
1
answer
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A Grothendieck fibration over a weakly contractible category with weakly contractible fibers is weakly contractible?
Let $B$ be a weakly contractible category (that is, it has a weakly contractible nerve), and let $F:E\to B$ be a Grothendieck fibration. Suppose further that the ordinary fibers of the Grothendieck ...
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12
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0
answers
772
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What's so difficult about $\pi_{15}(SO)$?
Regarding the table of $SO(n)$s-of-origin in Davis+Mahowald (if you can get MathSciNet), is there a good reason that it should take longer for $\pi_{15}(SO)$ to be representable than $\pi_{19}(SO)$, ...
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6
votes
0
answers
603
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Fiber sequences in proper model categories
I am confused about the notion of a fiber sequence (or dually a cofiber sequence) in a general pointed and proper model category $\mathcal{C}$.
Following Hovey, we can define, like in topology, a map ...
- 61
5
votes
2
answers
592
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Duality between proper homotopy theory and strong shape theory
In the n-lab entry about shape theory one can read that
Strong Shape Theory [...] has, especially
in the approach pioneered by Edwards
and Hastings, strong links to proper
homotopy theory. ...
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4
votes
0
answers
1k
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When are "diagrams of cofibrations" projectively cofibrant?
Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a diagram of cofibrations if for every object $p\in P$, $F(p)$ is ...
- 19.4k
2
votes
1
answer
329
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truncation commutes with localization?
Suppose I have a higher category, say a simplicial category $C$, and I want to invert a certain type of morphism $W$. Simplicial localization (for example hammock localization), gives a localized ...
- 807
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votes
2
answers
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The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?
Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...
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3
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0
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Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor
Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
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3
votes
2
answers
454
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Nonabelian cohomology via crossed modules
Let $G$, $H$ be topological groups and let $t \colon G \to H$ be a homomorphism, such that $t \colon G \to H$ is a topological crossed module. For a topological space $X$ we can define the nonabelian ...
- 7,537
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1
answer
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Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?
Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
- 19.4k
21
votes
2
answers
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obstruction theories in algebraic geometry
I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...
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votes
2
answers
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Proof of Bott Periodicity in twisted K-theory
I have a question about the Proof of Bott Periodicity in twisted K-theory
by Atiyah and Segal in their paper Twisted K-theory.
Following their notation, to prove Bott periodicity in this context it ...
7
votes
1
answer
1k
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categorifying induction in homotopy type theory
In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts.
Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I'd like to think of ...
- 5,535
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votes
1
answer
538
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Does adding degeneracies to a semi-simplicial diagram change the homotopy colimit?
Let $\Delta_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta_{+} \hookrightarrow \...
- 3,887
18
votes
4
answers
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Spaces that are both homotopically and cohomologically finite
Is it true that every connected space with
1) just finitely many nontrivial homotopy groups, all finite,
and
2) just finitely many nontrivial rational cohomology groups, all finite rank,
is ...
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5
votes
2
answers
445
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Which $H$--groups satisfy the rigidity property of abelian varieties?
Let us call a group object $G$ in a category $\mathcal C$ rigid, if it has the following property: For every group object $X$ in $\mathcal C$, every morphism $G\to X$ in $\mathcal C$ respecting the ...
- 3,975
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1
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676
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homotopy transfer for sheaves of algebras
homotopy transfer for algebras
Let $A$ be a differential graded (dg) $k$-algebra, and $H(A)$ its cohomology. $H(A)$ is naturally equipped with the structure of a graded algebra. In general we don't ...
- 8,113
8
votes
1
answer
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Construct a CW complex with prescribed homotopy groups and actions of $\pi_1$.
How to construct a CW complex $X$ with prescribed homotopy groups $\pi_i(X)$ and prescribed actions of $\pi_1(X)$ on the $\pi_i(X)$'s?
- 1,672
7
votes
2
answers
875
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Uniqueness of loop spaces
Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$.
Under what assumptions is (the homotopy type of) $Y$ unique?
As has been pointed out below, the ...
- 1,160
2
votes
1
answer
211
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Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
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41
votes
2
answers
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Homotopy groups of $S^2$
in the paper
Foundations of the theory of bounded cohomology,
by N.V. Ivanov, the author considers the complex of bounded singular cochains on a simply connected CW-complex $X$, and constructs a ...
- 3,719
2
votes
1
answer
779
views
weak equivalence of simplicial sets
Given a morphism f:X --> Y in sSet, and assume that it induces isomorphisms for \pi_0,\pi_1,\pi_2, and all integral homology groups. Does it imply that f is a weak equivalence?
In Hatcher's Algebraic ...
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3
votes
1
answer
470
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Parallelizability of exotic structure
I'd like to discuss a little bit about the problem I asked Diarmuid Crowley that whether all smooth structures on $S^7$ are parallelizable. He first came to using semi-characteristic but then came up ...
- 883
3
votes
2
answers
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References for Grothendieck's "Pursuing Stacks" and "Esquisse"
Where can I find Grothendieck's "Pursuing Stacks"/"A la poursuite des champs" and "Esquisse d'un programme"?
- 911
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0
answers
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Functor that maps to both $KO^n$ and $KO^{-n}$
(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...
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14
votes
2
answers
1k
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Homotopy type of Hilbert schemes of points of $\mathbb C^2$
Let $X=\mathbb C^2$, let $X^{[n]}$ be the Hilbert scheme of length $n$ 0-cycles in $X$, and let $X^{[n]}_0$ be the closed subscheme formed by the 0-cycles supported at 0. As far as I know $X^{[n]}_0$ ...
- 143
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votes
8
answers
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Examples of Brown (co)fibration categories that are not Quillen model categories?
K.S. Brown has shown that much of abstract homotopy theory can be carried out in the setting of Brown (co)fibration categories [MR0341469]. The decisive property, immediate from the axioms, is that ...
5
votes
1
answer
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Why are connective spectra called "connective"?
Recall that a spectrum is called connective if it is $(-1)$-connected (that is, its homotopy is concentrated in nonnegative degrees).
However, this left me scratching my head a bit. Why "connective"?...
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3
answers
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What is the homotopy theory of categories?
I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of ...
- 25.3k
6
votes
1
answer
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Geometric meaning of torsion in homotopy groups
It is not too hard to understand the geometric meaning of torsion in homology groups of CW complexes. However, I thought it would be interesting to hear how people describe/think of the geometric ...
- 2,015
2
votes
1
answer
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Reference for an automorphism in a paper of Toda
In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".
Unfortunately, Selick ...
- 12.5k
4
votes
1
answer
795
views
Complex orientation of the Adams Summand
First lets fix a prime $p$ (I really care about $p=2$ but would be happy to know about other primes as well). When localized at a prime the spectrum $ku$ (Complex connective K-theory) splits as a ...
- 3,676
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votes
1
answer
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Is the $\infty$-category of stable $\infty$-categories stable?
More generally, are there any remarkable properties enjoyed by the $\infty$-category of stable $\infty$-categories?
- 133
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votes
1
answer
910
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Classifying spaces of E_1 - spaces
Hello,
I try to understand aspects of homotopy coherence, in particular "recognition principle" of May.
About the following I did not think a lot, but I decided to ask here anyway, so to save ...
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votes
6
answers
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Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets ...
- 25.3k
7
votes
1
answer
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Homology of homotopy fixed point spaces
This is a general question for the homotopy theory crowd: How does one go about computing the homology and homotopy groups of homotopy fixed point spaces $X^{hG}:= Map^G(EG, X)$ for the action of a ...
- 4,083
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votes
2
answers
591
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Can a smooth, immersed loop in R^2 become not nullhomotopic by removing a point?
ATT
More precisely, let $\gamma :S^1\rightarrow R^2$ be a smooth immersed loop, the question is whether it is true that there is a point $p\in R^2-\gamma(S^1)$ such that $\gamma$ is not homotopic to ...
- 1,079
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votes
1
answer
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Slick Proof of Kudo Transgression Theorem
The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example,...
- 12.5k
5
votes
1
answer
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Flat Principal Connections and Homotopy Groups?
I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...
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3
votes
1
answer
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What is a left dual up to homotopy?
My question is prompted by
57589
If $X$ is an object in a monoidal category with unit $I$ then $Y$ is a left dual
if we have $I\rightarrow Y\otimes X$ and $X\otimes Y\rightarrow I$ which satisfy
the ...
- 9,052
17
votes
4
answers
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Degeneracies for semi-simplicial Kan complexes
By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. ...
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2
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Classifying space for S1-bundle?
What is the classifying space for $S^1$-bundle? Here, $S^1$-bundle means a fiber bundle which doesn't mean that it is principal $S^1$-bundle.
I know that for a space $F$,
$\lbrace$the set of fiber ...
- 115
10
votes
1
answer
911
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Analogue to Serre spectral sequence for cofiber sequences and homotopy
(This is a follow-up question to this one).
As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well ...
- 2,712
7
votes
2
answers
831
views
What are the algebras over $\Omega^k\Sigma^k$ ?
Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair
$$
\Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k,
$$
where $\Sigma^k$ is the $k$-th supension functor and $\...
- 101
6
votes
1
answer
417
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Homotopy dimension of a mapping
The homotopy dimension $h \dim X$ of a space $X$ is the minimal dimension of a CW-complex $Z$ homotopy equivalent to $X$.
I am interested in the generalisation to maps $f\colon X\to Y$. Here's what I ...
- 35k
7
votes
1
answer
436
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How equivalent are the theories of reduced and groupal $\infty$-groupoids?
I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...
- 53.1k
7
votes
2
answers
1k
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homotopy pushout of spaces homotopic to finite CW complexes
Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW ...
- 71
2
votes
1
answer
326
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Model categories and cellular maps
A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
- 3,676