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.
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Map between homotopy groups of O, related to J-homomorphism and K-theory of Z
Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi_{8s+1}^s = \pi_{8s+1} ((B\...
39
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What can topological modular forms do for number theory?
Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...
9
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What's the detailed proof of "the composition of planar tangles is well-defined"?
In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...
2
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2
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Convergence of a sum with the ranks of homotopy groups
Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...
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stable splitting into a wedge sum [closed]
Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee _{k=1}^\...
7
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0
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318
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Homotopy pullback preserving functor
In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy ...
12
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1
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732
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The multiplication on $THH$ of finite fields
Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...
14
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1
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650
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Alexander duality for non-manifolds
Let $X$ be a CW complex and $A$ a subcomplex. I will assume that both are compact, and that $X$ is $n$-dimensional. Furthermore, assume that the local homology of $X$ is that of a manifold in a ...
5
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0
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is there a moduli of stable infinity categories?
I know there exists a groupoid-valued prestack parameterizing (connected) triangulated dg-categories (ie the points of this moduli are not objects of a fixed dg-category, but rather dg-categories ...
5
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Inverse limit in shape theory
Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...
4
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Model structure on non-negative differential graded algebras with homological grading
I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of non-...
12
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2
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551
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Why does $Mf$ always support an $Mf$-orientation?
Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...
15
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1
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Homotopy transfer in the opposite direction
Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
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3
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When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$
For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...
4
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2
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582
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The classifying space of an infinite totally ordered set is contractible
I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...
4
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0
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Fundamental groups of stably parallelizable manifolds
Is it possible to realize every finitely presented solvable group as a fundamental group of a stably parallelizable closed n-manifold? If not, are there any known counterexamples?
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Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces
Warmup:
Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...
0
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1
answer
764
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unordered configuration space of pointed space
Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid \forall i\neq j: x_i\neq x_j, \}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an ...
3
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2
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590
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How to show the following two definitions of homotopy monomorphism are equivalent?
Let $M$ be a model category. In Toen's The homotopy theory of dg-categories and derived Morita theory Page 11 it is written:
a morphism $x \to y$ in a model category $M$ is called a homotopy ...
3
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Is the suspension of a weak equivalence again a weak equivalence?
Of course, the answer to this question depends on what we mean by suspension. If we work with based spaces and take the reduced suspension, the answer seems to be NO:
Take $X = \mathbb N$ (a ...
1
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1
answer
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Unordered configuration space of $\mathbb{R}P^1$
In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ...
3
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1
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349
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Do there exist nontrivial motivic cohomology operations preserving weights?
Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...
3
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0
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214
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Homotopy (co)limit (co)cones
Let $\mathscr{M}$ be a model category and let $\mathscr{I}$ be a small category. Consider any homotopy colimit functor $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}\colon\mathscr{M}^{\mathscr{I}}\...
2
votes
1
answer
234
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$\pi_0$ of a cosimplicial space
Let $n\mapsto X^n$ be a cosimplicial simplicial set and $X:= \underset{\longleftarrow}{\rm holim}\ X^n$ the homotopy limit. Is the natural map
$$ \pi_0(X) \to \underset{\longleftarrow}{\rm lim}\ \...
18
votes
1
answer
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What if homotopy were expanded to allow any connected space instead of $[0,1]$?
What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than $[0,1]$?
Given continuous $f,g:X\to Y$, define $f$ and $g$ to be C-...
7
votes
2
answers
632
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Direct proof that the model category of cdgas is left proper
Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
10
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1
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560
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Dyer-Lashof operations and the homology of GL_n
For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...
2
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1
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735
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Classifying space of a colimit of topological categories
Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...
35
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2
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
6
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1
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413
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When are principal bundles preserved by colimits?
Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...
3
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2
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435
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Example s.t. the unbased loop-space is not $\Omega X \times X$
For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...
5
votes
2
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561
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Maps to the group completion
Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary):
What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...
2
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0
answers
320
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Groups acting on complexes
Let $G$ be a finite group. We define a $G$-simplicial complex $\mathcal{A}(G)$ with set of vertices $G^*:=G-\{e\}$ and the simplices are the abelian subsets of $G^*$. The groupe $G$ act simplicially ...
13
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1
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Cohomology of the image of J spectrum
Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e.
$$ [j, HZ/p]_*$$
as a module over Steenrod algebra. ...
4
votes
1
answer
429
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Is the infinity-groupoid of a finite CW complex finitely-presented?
An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions.
Is the infinity-groupoid of a finite ...
8
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1
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603
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Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces
In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
There exist $\Sigma$-free operads $\mathcal{C}...
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0
answers
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Explicit calculation of G-CW(V) structure of a G-space
I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...
1
vote
1
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Homotopy invariance of Kan nerve of simplicial categories
The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the ...
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0
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Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR
Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
...
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A construction with homotopy colimits and homotopy pullbacks for descent
EDIT: Following the lines of some suggestions in the comments below, I try to add something more to explain the problem better. A map $\text{hocolim}Y\rightarrow\bar{Y}$ in $\text{Ho}(\mathbf{M})$ is ...
4
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1
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Glueing a property via homotopy colimits
I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf .
In the proof of Lemma 2.11, we are given a ...
4
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1
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Homotopical categories, the 2-out-of-6 property, and saturation
A homotopical category is a cateogry with a class $\mathcal W$ of arrows called weak equivalences which satisfies the 2-out-of-6 property.
The nlab article shows a deep connection between $\mathcal W$...
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Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?
I asked this question on math.stackexchange a week ago, but did not get an answer.
First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...
6
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1
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When does the Borel construction have the homotopy type of a CW-complex?
Suppose that $G$ is a Lie group acting smoothly on a manifold $M,$ does the Borel $M \times_G EG$ construction have the homotopy type of a CW-complex? If not, under what conditions would this be true? ...
2
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0
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Why is the oriented $G$-homotopy type of a $G$-complex uniquely determined by the periodicity generator?
Say we have a periodicity generator $e \in H^k(BG)$. I can show that we then have a $(k-1)$-dimensional $G$-complex $X$ with free $G$-action. It's also not that difficult to see that it has trivial $G$...
12
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3
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When are (weak) homotopy equivalence testable on open covers?
I asked this question on math.stackexchange, but did not get an answer.
Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't ...
3
votes
1
answer
361
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cohomology ring of configuration spaces
In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...
6
votes
2
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641
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homology of a mapping spectrum
If $X$ and $Y$ are two spectra, I denote by $F(X,Y)$ their mapping spectrum. This is uniquely determined by the existence of a natural isomorphism $[X\wedge Y, Z]\cong [X,F(Y,Z)]$.
I denote by $H_*$ ...
0
votes
0
answers
52
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Criterion for projectively cofibrant diagrams? [duplicate]
(repost from math.stackexchange)
Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The ...
44
votes
1
answer
8k
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Homotopy Type Theory: What is it?
My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following:
There are three directions:
...