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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Is every simplicial map $\Phi:K(A, n) \to K(A', n)$ a simplicial homomorphism of groups?
I have posted a few questions on MSE, most notably this one, which revolve around the same issue and have received no answers, so I decided to ask the same here.
In the following, $K(A, n)$ is the ...
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Are these CW-complexes homotopy dominated by each other?
Let $K_i$ for $i$ a prime, be the 2-dimensional CW-complex with a single vertex associated with the presentation $$\mathcal{P}_i=\langle r,s,t\; |\; s^2 =t^3, [r^2 ,s^{2i+1}]=1,[r^2 ,t^{3i+1}]=1\...
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Grothendieck spectral sequence and exact couples
I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges ...
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Spanier-Whitehead dual of space of natural transformations
Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra).
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How to get by with only functorial cylindrical objects?
In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects /...
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Homotopy coherent generalization of classifying space theory
Classically, given a compact Lie group $G$, there is a topological space $BG$ which classifies principal $G$-bundles. This means that there is an equality of sets {principal $G$-bundles up to ...
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A $p$-adic homotopy theory for non-simply connected spaces?
I'm looking to understand the state of the art for $p$-adic (unstable) homotopy theory of non-simply connected (non-nilpotent!) spaces. Ideally, I'd also like integral versions, e.g. things like ...
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Poincaré series of deloopings of finite complexes
Recall that the Poincaré series of a topological space $X$ is defined as $P_X(t) = \sum_{j=0}^{\infty} b_jt^j$, where $b_j = \text{dim}_{\mathbb Q} H_{j}(X;\mathbb Q)$ means the $j^{\text{th}}$ (...
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Fundamental group of intersection of two codimension=2 complements
$X$ is a smooth manifold. $Y_1\subset X$ and $Y_2\subset X$ are both complements of a finite collection of real codimension=2 (transversal intersecting) submanifolds. Suppose $Y:=Y_1 \cap Y_2$ is ...
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Kunneth formula for topological K theory
Atiyah proved the Kunneth formula for topological K theory for finite CW complexes.
Does Kunneth formula hold as spectra? That is, for finite CW complexes $S_1,S_2$, is there a weak equivalence
$$
K(...
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CW-complex $X$ has the homotopy type of a finite wedge of spheres
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
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How now to study operads in homotopy theory?
There is a great introduction by May, "The Geometry of Iterated Loop Space". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to the ...
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Dependence of completion on the base ring
Let $M$ be a module over an $E_\infty$ ring, $A$. Let $I$ be an $A$-non unital commutative algebra together with an associative map $I \wedge_A M \to M$.
Define ${_A}(M/I^n)$ as the cofiber of $I^{\...
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Numerator in the zeta values at negative odd integers
The real J-homomorphism produces cyclic subgroups of size the denominator of $\zeta(1-2k)=-B_{2k}/k$ for $k>0$ in $\pi_{4k-1}S$ which completely account for first layer of the chromatic filtration ...
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Can every finitely presented group be realized as a fundamental group of a compact four-dimensional smooth submanifold $\mathbb{R}^4$?
Every finitely presented group is realized as a fundamental group of a two-dimensional complex (a simple exercise on Van Kampen's theorem). I was told that a two-dimensional complex can be well ...
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Bernoulli numbers and the chromatic filtration on the stable homotopy groups of spheres
It is well known [Clausen, p-adic J-homom., in introduction] that there are cyclic subgroups of $\pi_{4k-1}S \: (k>o)$ with size the zeta values $B_{2k}/k \: (=-\zeta(1-2k))$ which completely ...
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How to calculate the periodic cyclic homology group of $\overline{\mathbb{Z}}/\mathbb{Z}$
$\newcommand{\ur}{\mathrm{ur}}$Fix a prime number $p$. We let $\overline{\mathbb{Z}}$ denote the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Z}}_p$ denote the ...
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Barycentric subdivision and 1-coskeletalization
Let
$sd : sSet \to sSet$ denote barycentric subdivsion;
$cosk_1 : sSet \to sSet$ denote 1-coskeletalization.
Question: Let $X$ be a graph or simplicial set. If the homotopy type of $cosk_1(X)$ is ...
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Do elements of every order occur in homotopy groups of spheres?
It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?
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Why the Bousfield localization of spectra at topological K group is important?
Recently, Akhil Mathew has published papers on $K(1)$-local theory:
On $K(1)$-local $\mathrm{TR}$ and
Remarks on $K(1)$-local $K$-theory.
What is the motivation of $K(1)$-local theory?
What does $K(1)$...
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Chromatic filtration on stable homotopy
Chapter 5 of Ravenel's green book starts with the sentence “[The chromatic spectral sequence] is a mechanism for organizing the Adams-Novikov $E_2$ term and ultimately $\pi_*(S^0)$ itself." My ...
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Classifying cohomology with local coefficients
Is there a low level "homotopical" description of cohomology with local coefficients? Similar to the identification of ordinary singular cohomology with the homotopy classes of maps to the ...
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Bousfield class of $TMF$ and $E(2)$
Let us work concretely at the prime 3. How does $TMF \wedge X \simeq E(2)$ for X a finite type 0 spectrum imply that $TMF$ and $E(2)$ have the same Bousfield class?
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Is the composite of absolute derived functors a derived functor?
Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A total left ...
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A functor admitting a total, but not point-set derived functor
Mike Shulman, in his article Homotopy limits and colimits and enriched homotopy theory observes (towards the end of page 8) that it may be the case that a functor $F: C \to D$ between homotopical ...
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Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
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Can we define derived functors in model categories without functorial factorisations?
Let $F: \mathcal{C} \to \mathcal{D}$ be a left Quillen functor between model categories. In Definition 2.16 of Goerss–Schemmerhorn - Model Categories and Simplicial Methods, the left derived functor $...
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Constructing Postnikov base functor from Brown-like representability
I am looking through the paper "On the Representability of Homotopy Functors" by Heller. At the start of Section 4, he considers the homotopy category $H(n)$ of $n$-truncated pointed ...
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(Non-)Completeness for connected pointed $\infty$-groupoids
In an example in Lurie's HA, it is implied that the $\left(\infty,1\right)$-category of connected pointed $\infty$-groupoids is presentable. But it is not closed under homotopy pullbacks (e.g., $\...
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The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
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Homotopy coherent nerve versus simplicial nerve
Background
Recently I asked a question on a particular construction ot the classifying space of a topological group. I got an answer, but it relied on nontrivial Quillen equivalences between various ...
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When does Tate spectral sequence degenerate at $E_2$?
For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
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When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
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Some details about a proposition of Wall
Let $X$ be a connected CW-complex. For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi}...
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Preservation of fiberwise normal bundles under fiberwise homotopy equivalences
I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.
A ...
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Simplicial nerve of a topological group
Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric ...
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Configurations of points in a spectrum
I am wondering if the following construction has appeared in the literature:
Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \...
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Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?
Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...
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Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 March 2022)
In Theorem 2 of these notes, Ralph Cohen reformulates the main theorem of Hirsch-Smale theory merely in terms of normal bundles.
In particular, he says that if $N, M$ are two manifolds, $\dim N< ...
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Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?
In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy ...
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Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?
Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
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On the proof of the surgery step in Wall's book
This question regards a part of the proof of the so called surgery step, in Wall's book "surgery on compact manifolds", Theorem 1.1.
Setting
$M^m$ smooth manifold, $X$ CW complex, $\phi :M\...
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What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
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About the equivariant analogue of $G_n/O_n$
Let $BO_n$ and $BG_n$ be the classifying spaces for rank $n$ vector bundles and for spherical fibrations with fiber $S^{n-1}$, respectively, and let $G_n/O_n$ be the homotopy fiber of $BO_n\to BG_n$.
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When did the Joyal model structure on simplicial sets originate?
Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...
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Timeline of "foundational" advances in homotopy theory?
As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been ...
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Homotopy limits indexed by a covering
We all know that the homotopy fiber of a continuous function $f: A \to B$ has a simple expression in terms of points and paths connecting them, that is
$$ \textrm{hofib}_y(f) = \{ (x,\gamma) \in A \...
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Mark Hovey's open problems in the theory of model categories
Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not.
My question is:
i) which of the 13 ...
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Under which conditions is the bar construction a conservative functor?
The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends ...
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The mapping cylinder of a map between spaces that are homotopy equivalent to CW complexes
Suppose $X$ and $Y$ are spaces that are homotopy equvialent to CW complexes, and let $f:X\to Y$ be a continuous map. I am trying to show that the pair $(M_f,X)$ is homotopy equivalent to a CW pair.
I'...
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