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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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?
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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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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 ...
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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 ...
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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-...
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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 ...
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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 (...
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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 (...
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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 ...
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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}]...
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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 ...
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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 ...
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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....
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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 ...
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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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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^...
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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 "...
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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 $\Phi: \lambda \to \mathrm{CGWH}$...
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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 (...
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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 ...
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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 $i$-...
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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) $k$-...
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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 $A_\infty$ ...
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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 ...
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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 ...
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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 $S: \mathsf{C} \to \mathsf{C}$...
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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 $\bigcirc_B X$ ...
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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 ...
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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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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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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 ...
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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 $DM_k$...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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"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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $H\mathbb{Z}$-...
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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 ...
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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 ...
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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:\...
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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 ...
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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 ...
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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 ...