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.

Filter by
Sorted by
Tagged with
7 votes
1 answer
2k views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite limits? ...
Ulrich Pennig's user avatar
7 votes
1 answer
597 views

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 ...
Ulrich Pennig's user avatar
7 votes
1 answer
451 views

The geometric meaning of the higher quotient by the commutant ideal

The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal. For any dg-algebra $A$ let $A_{Ab}$ denote the derived ...
nikitamarkarian's user avatar
7 votes
1 answer
915 views

cosimplicial algebras to dg-algebras

The normalized Moore complex functor is usually considered taking simplicial abelian groups to chain complexes. But there is an obvious dual version that takes cosimplicial abelian groups to N-graded ...
Urs Schreiber's user avatar
7 votes
1 answer
278 views

Can you construct a mapping space from local data? (looking for reference)

I'd to know if/where there is a reference for the following construction. Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
Kevin Walker's user avatar
  • 12.3k
7 votes
1 answer
151 views

Reference request for equivalences between different models of lax limits

There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
happymath's user avatar
  • 167
7 votes
1 answer
433 views

Geometric realization of a poset

Consider the finite Boolean lattice $B_n$ of subsets of $[n]:=\lbrace 1,\dots,n\rbrace$ ordered by inclusion, let $1\leq j,k\leq n$ and consider the poset: $$A_{j,k}=\lbrace\emptyset\neq U\in B_n\mid (...
Marcos's user avatar
  • 577
7 votes
1 answer
272 views

Homotopy fixed points of involutive automorphisms of discrete groups

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (...
rvk's user avatar
  • 553
7 votes
2 answers
202 views

Cofibrancy of a right module over an operad

If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module ...
Connor Malin's user avatar
  • 5,191
7 votes
1 answer
477 views

Categorical Significance of Fibrations

It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
Ronald J. Zallman's user avatar
7 votes
1 answer
862 views

Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
user avatar
7 votes
1 answer
311 views

Are these two notions of unstable localization suitably equivalent?

It seems to me that although homological localization (i.e. formally inverting $E$-homology equivalences for some $E$) is a reasonable thing to do to a spectrum, it's a pretty brutal thing to do to a ...
Tim Campion's user avatar
  • 60.6k
7 votes
1 answer
306 views

Does a filtered A_N algebra give rise to a multiplicative spectral sequence?

The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728....
algebrachallenged's user avatar
7 votes
2 answers
452 views

Components of a loop space, semidirect products, and multiplicativity

Let $(X, x_0)$ be a based topological space, and $\Omega X$ its based loop space. The group of path components of $\Omega X$ is $\pi_0(\Omega X) = \pi_1(X, x_0)$. For brevity, let's call this group ...
Craig Westerland's user avatar
7 votes
1 answer
483 views

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 ...
David White's user avatar
  • 29.4k
7 votes
1 answer
708 views

Hopf fibration inside the retraction of R^4 minus line -> S^2?

This was inspired by this question. Let $Y = {\mathbb R}^4 \setminus$a coordinate line, which retracts to ${\mathbb R}^3 \setminus$a point, which retracts to $S^2$. What is an explicit immersion $S^...
Allen Knutson's user avatar
7 votes
1 answer
410 views

Different definitions of homotopy colimits

I was reading about the definition of homotopy limits and colimits, and I have seen two different approaches in "Homotopy theories and model categories" by Dwyer and Spalinski, and in "...
MikeTrooper's user avatar
7 votes
1 answer
200 views

Quasifibrations and transfinite filtrations

This question takes place in the category $\mathrm{CGWH}$ of compactly generated weak Hausdorff spaces. Let $\lambda$ be a limit ordinal, and suppose we have a diagram $\Phi: \lambda \to \mathrm{CGWH}$...
Jeff Strom's user avatar
  • 12.5k
7 votes
1 answer
216 views

Terminology: "left homotopical"?

I first asked this on StackExchange, but no dice; so apologies in advance if this question really belongs there. Suppose a functor $F \colon \mathcal{C} \to \mathcal{D}$ between two model categories (...
prefix.crm114's user avatar
7 votes
1 answer
376 views

Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
user101010's user avatar
  • 5,319
7 votes
1 answer
229 views

Tits building of free modules

The Tits building for a vector space $V$ denoted by $T(V)$ is defined as a simplicial complex whose vertices are non-zero proper sub-vector spaces and edges are inclusion of subspaces and $i$-...
user127776's user avatar
  • 5,831
7 votes
1 answer
406 views

Construction for algebras over little cubes operad

Recently I came across the following construction: Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-...
Tashi Walde's user avatar
7 votes
2 answers
1k views

Deformation theory of co-$A_\infty$ structures.

The following question is related to my previous post on co-$A_\infty$ spaces (co-$A_\infty$ spaces), but goes in a somewhat different direction. Some Background: In trying to classify $A_\infty$ ...
John Klein's user avatar
  • 18.6k
7 votes
0 answers
176 views

Complex cobordism and integrable systems

In Jack Morava's paper On the complex cobordism ring as a Fock representation, it was remarked right at the beginning that complex cobordism may play a role in the theory of integrable systems. In ...
user1271629's user avatar
7 votes
0 answers
141 views

Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
Yuji Tachikawa's user avatar
7 votes
0 answers
264 views

Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
ಠ_ಠ's user avatar
  • 5,933
7 votes
0 answers
166 views

How does nullification of $K(\mathbb{Z}, 2)$ compare to 1-truncation?

Let $B$ be a type (or space). A type (or space) is $B$-null if the canonical map $X \to X^B$ is an equivalence. The $B$-nullification of an arbitrary type $X$ is a $B$-null type $\bigcirc_B X$ ...
aws's user avatar
  • 3,836
7 votes
0 answers
160 views

Relative version of Browder's theorem on H-spaces

A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's ...
Danny Ruberman's user avatar
7 votes
0 answers
156 views

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 $j^{\text{th}}$ (...
Jens Reinhold's user avatar
7 votes
0 answers
288 views

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 ...
Arshak Aivazian's user avatar
7 votes
0 answers
174 views

THH and string topology

There is an equivalence $THH(S^{\infty}_+ LX) = S^{\infty}_+ FX$ where $FX$ is the free loop space (I already used the letter $L$). The circle action on $THH$ gives rise to a degree $1$ cyclic ...
taf's user avatar
  • 448
7 votes
0 answers
447 views

The conservativity conjecture for motives

Let $k$ be a field of characteristic zero and $DM_k$ be the derived category of rational Voevodsky motives. As I understand, there are conjectures which state that there is a $t$ structure on $DM_k$...
user1092847's user avatar
  • 1,327
7 votes
0 answers
388 views

Transfer of E-infinity algebra structures

Skip to the bottom for my questions, first some discussion: It is a celebrated theorem of Kadeišvili that $A_{\infty}$-algebra structures can be transferred along homotopy equivalences so that the ...
J Cameron's user avatar
  • 551
7 votes
0 answers
226 views

What is the group completion of the groupoid of even finite sets and even permutations?

$\newcommand{\sgn}{\mathrm{sgn}}\newcommand{\defeq}{\overset{\mathrm{def}}{=}}$The Barratt–Priddy–Quillen–Segal theorem says that the $\mathbb{E}_\infty$-group completion of the groupoid of finite ...
Emily's user avatar
  • 10.3k
7 votes
0 answers
154 views

Homotopy equivalent cartesian product of closed manifold

I'm little bit lost with the following question: I have four connected closed orientable manifolds $M,N,S,S'$ such that $S$ and $S'$ are homotopy equivalent and $M\times S$ is homotopy equivalent to $...
Paris's user avatar
  • 707
7 votes
0 answers
209 views

The homotopy monoids of directed spheres

In directed homotopy theory, one replaces spheres by directed spheres and homotopy groups by homotopy monoids. Is it known what are the first few homotopy monoids of directed spheres? Do homotopy ...
Emily's user avatar
  • 10.3k
7 votes
0 answers
127 views

Endofunctors of the surface category

Let $\mathrm{Cob}_2$ be the symmetric monoidal $(\infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are ...
Jan Steinebrunner's user avatar
7 votes
0 answers
179 views

"Relative Whitehead products"

The notion of a relative Whitehead product exists in the literature and has been asked about before (e.g. here). I am trying to find out about a different product on relative homotopy classes which ...
Matt's user avatar
  • 198
7 votes
0 answers
199 views

Higher categorical / operadic approach to homotopy associative, homotopy commutative, $H_\infty$ ring spectra?

Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces ...
Tim Campion's user avatar
  • 60.6k
7 votes
0 answers
190 views

2d TQFTs with values in simplicial sets and Reedy categories

Let $Cob$ be the category such that $Obj(Cob)$ is $\emptyset\sqcup\mathbb{N}$, with $n$ seen as the union of $(n+1)$ circles numbered from $0$ to $n$, morphisms are (homeomorphism classes of ...
Sergei Burkin's user avatar
7 votes
0 answers
201 views

Does the Whitehead torsion of a homotopy equivalence depend on the CW structure?

In the (old) literature I've seen referenced the question of whether simple homotopy equivalence is a topological property, i.e. whether it depends only on the underlying space, rather than the ...
Connor Malin's user avatar
  • 5,191
7 votes
0 answers
174 views

The space of all constructions of $(\infty, 1)$-categories

In the paper How to glue derived categories the following is written about $(\infty, 1)$-categories: This idea was discussed quite a lot in the early 1990-ies, and probably before that, but people ...
jp86's user avatar
  • 71
7 votes
0 answers
314 views

THH of the tensor product of E_1 rings

Is there a formula or a spectral sequence relating the $THH$ of two $E_1$-rings and the $THH$ of their tensor product over $\mathbb{S}$? Krause and Nikolaus state without context on page 3 of their ...
user134312's user avatar
7 votes
0 answers
247 views

Homotopy invariant analogues of localizing invariants

Given a localizing invariant, $E$, valued in spectra, by following the recipe prescribed in 3.13 of https://arxiv.org/abs/1808.05559, we can define a homotopy-invariant version of $E$ on $H\mathbb{Z}$-...
Liam Keenan's user avatar
7 votes
0 answers
283 views

Derived symmetric powers and determinants

Given a vector bundle $V$ (on a scheme $X$, say), I can form $Sym(V[1])$, the symmetric algebra (in the derived/graded sense) on the shift of $V$; in other words this is the Koszul complex of the zero ...
Tom Bachmann's user avatar
  • 1,951
7 votes
0 answers
219 views

Homological and homotopical equivalence of complex analytic varieties

Consider a map between two complex analytic varieties of finite type $f:X\to Y$. Suppose that $f$ induces isomorphisms on cohomology with (constant) integral coefficients. Under what reasonable ...
W. Rether's user avatar
  • 395
7 votes
0 answers
270 views

Generalization of familiar theorem about singular homology to general model category

I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this? The second question is maybe related, I don't know. But anyway, given $U:\...
Noel Lundström's user avatar
7 votes
0 answers
270 views

Is it a coincidence that $Gal(\mathbb C / \mathbb R) \cong C_2 \cong Aut(E_1)$? (Or: why are $\mathbb C$-algebras with involution so useful?)

The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence ...
Tim Campion's user avatar
  • 60.6k
7 votes
0 answers
326 views

Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures? While I'm at it: is there an example of a tensor ...
Tim Campion's user avatar
  • 60.6k
7 votes
0 answers
220 views

Which spaces are most naturally presented simplicially?

It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward. In general, if I have a CW ...
Tim Campion's user avatar
  • 60.6k

1
24 25
26
27 28
60