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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A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
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Algebraic closure as a fibrant replacement?
Emil Artin's construction of the algebraic closure of a field $K$ is as follows. Let $K_{0} = K$, and inductively let $\{x_f\}$ be a set of indeterminates indexed by the irreducible $f$ in one ...
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What is the homotopy type of the space of the homeomorphisms of the n-ball whose restriction to the boundary is isotopic to the identity?
Consider the set of homeomorphisms of the topological n-ball to itself with the compact open topology. Sitting inside this space of homeomorphisms are particular subspaces. The first subspace is those ...
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Negative and periodic cyclic homology of a semi-free cdga
Let $A$ be a semi-free commutative differential graded algebra in non-negative degrees with differential $d$ of degree $-1$, over a field of characteristic zero. Recall that semi-free means that if ...
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Proper maps and transversality
I'll begin with the question, which is intrinsically interesting:
Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map $...
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Homotopical Galois theory of coverings
In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question.
I'm studying a little homotopical algebra in this article by Brown. You can easily ...
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Monoidal Model Categories with Suspension Functor
This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model ...
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Fubini theorem for hocolim
I wanted to ask the following question,
Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. ...
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The homotopy colimit of a tower of triangles
Set the framework to be a triangulated category with all set indexed coproducts.
In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268- 361, (...
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On the natural (bigraded) homotopy groups of a simplicial object in a model category
$\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- ...
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Second homotopy groups of 3-complexes and Fenn's spiders.
Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been ...
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deformation retraction of the complement
Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$. It is known that if $V$ is homotopy equivalent to $N$ then $X-V$ need not be homotopy equivalent to $X-N$, the Alexander horned ...
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Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
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Manifolds with prescribed fundamental group and finitely many trivial homotopy groups
Fix $G$, a finitely generated presented group.
It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with ...
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D(R) versus Ho(HR)?
Given an algebraic ring, how is its derived category related to the homotopy category of HR modules?
Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to ...
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Is $\mathbb{H}P^\infty_{(p)}$ an H-space?
Put $X=\mathbb{H}P^\infty$ (so $X$ classifies quaternionic line bundles, and $\Omega X=S^3$). There is no obvious reason for $X$ to be an H-space, because the tensor product of quaternionic vector ...
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How to endow an n-fold Segal Space with a symmetric monoidal structure?
I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: What is a symmetric monoidal $(\...
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K(r)-localization and monochromatic layers in the chromatic spectral sequence
While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...
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is $\pi_k(X_1,A_1)$ a direct summand of $\pi_{k}(X_1\vee X_2,A_1\vee A_2)$
For space pairs $(X_1,A_1)$ and $(X_2,A_2)$ ,is there a groups M,such that $\pi_{k}(X_1\vee X_2,A_1\vee A_2)\cong \pi_k(X_1,A_1)\oplus M$?
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Homotopy type of connected sum of certain manifolds
For closed n-manifolds M and N with the form $M\simeq M^{n-1}\cup _{f}e^n $,$N\simeq N^{n-1}\cup _{d}e^n$
Why $M\sharp N \simeq (M^{n-1}\vee N^{n-1})\cup _{f+d}e^n$?
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Does the following categorial sum preserve weak equivalences?
In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following category....
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$\mathbb{Z}/2$-action on spectra given by inversion
This is a somewhat naive question to which I don't know the answer. There is a map of spectra $S \to S$, defined up to homotopy, given by multiplication by $-1$, and it satisfies the relation $(-1)^2 \...
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Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?
Motivation
In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in $...
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Coherent MU_*-Modules
It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over $\Omega_\...
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Whitehead Theorem for Harmonic Spectra
What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite ...
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Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
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Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?
Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced ...
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Is homotopy definable by categorical means?
Is being homotopic - as a relation between two continuous functions, i.e. morphisms in Top - definable by categorical means? Can one detect from the context of dots and arrows, whether two parallel ...
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Does the Monoid Axiom hold for k-spaces?
In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness ...
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Topologically enriched homotopy colimits commuting with homotopy pullbacks
Hi,
I am looking for an enriched analogon of Proposition 4.4 in https://www.google.de/url?q=http://hopf.math.purdue.edu/Rezk-Schwede-Shipley/simplicial.pdf
Concretely, I would like to prove the ...
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Homotopy type of TOP(4)/PL(4)
It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
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Delooping and unreduced operads
Suppose I have an operad $P(n)$ in topological spaces in the sense of the book by Markl, Shnider and Stasheff, i.e. there is no space $P(0)$. In agreement with some of the answers to this question, I ...
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Two commuting operad actions
If the following is true, it is probably well-known to the experts. Nevertheless, I could not find a reference for it.
Suppose $P$ and $Q$ are $A_{\infty}$-operads in topological spaces and $X$ is a ...
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How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?
First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...
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How should one understand orbifold fundamental groups?
I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...
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Finiteness of stable homotopy groups of spheres
Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...
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Cohomology theory associated to the spectrum BG
Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf
Near the end of the text (p582) we ...
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Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions?
Let $G$ be a topological group and let $X$ and $Y$ be connected, well-pointed $G$-spaces. Suppose $f:X\to Y$ is a pointed homotopy equivalence and a $G$-equivariant map (but not an equivariant ...
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Localization at Infinite Wedges of K-theories or BP
This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges ...
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Homotopy theory of topological stacks/orbifolds
Motivation $\newcommand{\T}{\mathscr{T}}$
I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...
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Classifying spaces of topological groups that are not well-pointed
Let $G$ be a topological group.
The geometric bar construction $BG = B_{\bullet}(pt, G, pt)$ together with $EG = B_{\bullet}(pt,G,G)$ and the map $EG \to BG$ yields the universal principal $G$-bundle ...
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Homology-Cohomology Pairing
The Steenrod algebra $\mathcal{A}^*$ is the algebra of cohomology operations for the spectrum $\mathrm{H}\mathbb{F}_2$, i.e. $\mathcal{A}^*\simeq (\mathrm{H}\mathbb{F}_2)^*(\mathrm{H}\mathbb{F}_2)$.
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Is every long exact sequence of homotopy groups induced by a fibration?
Is every long exact sequence
$$\cdots\to\pi_{d+1}(B)\to\pi_d(F)\to\pi_d(E)\to\pi_d(B)\to\pi_{d-1}(F)\to\cdots$$
with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map ...
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smooth homotopy on exotic R^4
Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric.
Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to ...
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Is the geometric realisation of a nerve equivalent to the classifying space of its categorisation?
The nerve functor $N:Cat\to sSet$ has a left adjoint, namely the categorisation $C$. In fact there is a natural isomorphism $\epsilon: CN\to Id$ and $N$ is a full embedding.So if I start with a ...
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Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
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line bundles and universal covers
When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:
Suppose I have a (locally contractible) connected topological group $G$...
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Is there a sense in which the homotopy theory of simplicial sets is the "paradigmatic" one?
I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the category of ...
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Differentials in the Adams Spectral Sequence for spheres at the prime p=2
How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only ...
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3
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Second Homotopy Group of Graph Manifolds
A graph manifold is a closed 3-manifold $M$ that admits a finite collection of disjoint embedded tori $\mathcal{T}$ so that $M \setminus \mathcal{T}$ is a disjoint union of Seifert fibred spaces (i.e. ...
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