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.
2,477
questions
7
votes
0answers
95 views
Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum?
Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal ...
5
votes
1answer
185 views
Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$
We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv}...
4
votes
0answers
79 views
$\mathbb{E}_\infty$-refinements of the graded tensor product of $\mathbb{Z}$-graded spectra
The category
$$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$
of $\mathbb{Z}$-graded $R$-modules has a natural monoidal ...
2
votes
1answer
186 views
Corepresentability of involutory objects in monoidal $\infty$-categories
The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$).
A similar story ...
1
vote
0answers
129 views
Involutions in $\infty$-categories
$\newcommand{\id}{\mathrm{id}}$An involution in a category is a functor $\mathbf{B}\mathbb{Z}/2\to\mathcal{C}$, corresponding precisely to an object $X$ of $\mathcal{C}$ together with a $\mathbb{Z}/2$-...
1
vote
0answers
131 views
Homotopy section but no pointed homotopy section
Can anyone give an example of a pointed map $p:(E,e)\to (B,b)$ between connected pointed spaces (reasonably nice, say of the homotopy type of CW complexes) such that $p$ admits a homotopy section but ...
2
votes
1answer
121 views
A question about cofiber diagrams in stable $\infty$-categories
My question is as follows say I have a commutative diagram
$\require{AMScd}$
\begin{CD}
X @>f>> Y @>g>> Z\\
@V \alpha V V @VV \beta V @VV \gamma V\\
X’ @>>f’> Y @>>g’&...
1
vote
0answers
61 views
construction of $K_0$-group and Karoubian completion
Let $A$ be a ring. The $K_0$ group of $A$ can be defined in most
old fashioned way as the Grothendieck group of the set of isomorphism classes
of its finitely generated projective $R$ modules, ...
1
vote
1answer
202 views
Is there an operad homotopifying the Koszul rule?
In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...
1
vote
0answers
230 views
Applications of $\mathbb{Z}$-graded algebraic geometry to algebraic topology
There's a theory of algebraic geometry over $\mathbb{Z}_2$-graded commutative rings, often called "algebraic supergeometry" or the theory of superschemes. From what I understand, there's ...
5
votes
0answers
138 views
Generalising supercommutativity as a grading by the $1$-truncated sphere spectrum
A discussion that has been going recently is that supersymmetry corresponds to grading over the sphere spectrum, coming from an insight due to Kapranov.
To formalise such a statement, one needs a ...
4
votes
1answer
122 views
Interpolating between the flat and smooth affine lines in spectral algebraic geometry
Consider the following construction (which came up recently in a question about "spectral exterior algebras"):
Pick a ring spectrum $R$ and consider the $\infty$-category $\mathsf{Mod}_R$ ...
5
votes
2answers
540 views
Is there a "spectral exterior algebra" construction in higher algebra?
Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by
$$
\begin{align*}
\mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=...
2
votes
0answers
115 views
pseudo-abelian category / Karoubian category in K-theory
A pseudo-abelian category or Karoubian category $\mathcal{C}$ is a preaditive
category such that every idempotent morphism
$i: A \to A$ in $\mathcal{C}$ has a kernel and consequently a
cokernel as ...
6
votes
1answer
515 views
What is the homotopy category of the sphere spectrum?
Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?
6
votes
1answer
326 views
Grading ring spectra over the sphere spectrum
$\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in ...
1
vote
0answers
69 views
Homotopy type of complement to a union of linear subspaces
Im not sure if this question is appropriate for MO, but I'm looking for a hint about some questions about homotopy type of complement to a union of linear subspaces in vector space $\mathbb{R}^n, \...
2
votes
0answers
132 views
Relation between Bott-Samelson theorem and James reduced product
I asked this question on the homotopy theory chat, but I haven't got any answer - thus I decided to post it as a question here.
The question is rather historical. Let $X$ be a connected topological ...
24
votes
1answer
517 views
Are complex-oriented ring spectra determined by their formal group law?
To every complex-oriented ring spectrum $E$ there is associated a formal group law, which is a power series $F_E(x,y)\in E_*[[x,y]]$.
Suppose $E$ and $F$ are two complex-oriented ring spectra and ...
5
votes
0answers
137 views
(Co)homology of a directed space with coefficients in a commutative monoid
This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a ...
5
votes
0answers
80 views
Variations on Thomason's equivalence between connective spectra and symmetric monoidal categories
There's a number of results relating monoidal categories to connective spectra (which are themselves equivalent to $\mathbb{E}_{\infty}$-spaces):
Symmetric monoidal categories model all connective ...
11
votes
1answer
531 views
Is there a concrete description of the nontorsion elements in the homotopy groups of spheres?
By Serre's theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have ...
9
votes
0answers
182 views
Is an $n$-connected space a homotopy colimit of $n$-connected spheres?
A pointed, $n$-connected space $X$ admits a CW structure using just $n$-connected spheres, so $X$ is an iterated homotopy colimit of $n$-connected spheres (i.e. you start with spheres and keep taking ...
3
votes
1answer
163 views
What is the pointed Borel construction of the $0$-sphere?
From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $X\times_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category ...
3
votes
0answers
101 views
$(-n-1)$-connected spectra vs. reduced excisive functors from $n$-truncated pointed spaces
It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of ...
14
votes
1answer
281 views
On the homological dimension of a Borel construction
Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$...
4
votes
0answers
158 views
Weakening the excision condition for spectra
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently, I've noticed that the definitions of special $\...
3
votes
1answer
160 views
Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\...
3
votes
0answers
122 views
Rational mapping from affine space to projective space
This problem is highly related to this one. Over there I learnt about the Segal's problem regarding the rational mapping spaces. There seems to be a generalization of the result of the Segal for ...
6
votes
0answers
335 views
What do we know about the homotopy of $\mathrm{Q}S^0\wedge\mathrm{Q}S^0$?
The homotopy groups of $\mathrm{Q}S^0$ are the stable homotopy groups of spheres, of which we know the first 81 exactly, and the first 90 up to some uncertainties. A table is given in Wikipedia:
Do ...
3
votes
0answers
316 views
What do we know about $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and the spectral DM Stack $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$?
These days I've been trying to wrap my head around the current proposed approaches to algebraic geometry over the elusive "field with one element", one of whose main objects of interest is ...
1
vote
1answer
84 views
Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]
Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
2
votes
1answer
195 views
Are continuous maps generically homotopic to a regular map?
Let $X$ and $Y$ be smooth algebraic varieties over $\mathbb{C}$. Let $f$ be a continuous map from complex points of $X$ to $Y$. Are there Zariski opens $U$ and $V$ inside $X\times \mathbb{A}^1$ and $Y\...
3
votes
1answer
185 views
Reference request: functoriality of $\underline{E}$ and $\underline{B}$
For any group $G$, the universal example for proper $G$-actions, $\underline{E}G$, is a proper $G$-space such that for any other proper $G$-space $X$, there exists a map (unique up to $G$-equivariant ...
12
votes
1answer
282 views
Roadmap for L-Theory
Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ...
5
votes
2answers
292 views
Examples of $\mathbb{E}_{k}$-semiring spaces
Semirings, also called rigs, are rings without negatives: their underlying additive monoids are not groups (in other words, while rings are monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, ...
4
votes
0answers
244 views
Can one define fields in stable homotopy theory via invertibility?
In Nilpotence in Stable Homotopy Theory II, Hopkins–Smith define a field spectrum to be a ring spectrum $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$. They then show that any field ...
16
votes
0answers
240 views
Fibrations whose total spaces are more highly connected than their fibers
The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers.
Are ...
15
votes
3answers
838 views
Role of univalence in homotopy group calculations
This book has a section with proofs of the fact $\pi_1(S^1)=\mathbb Z$ using the univalence axiom. They are a bit too technical for me at the moment to read, but I want to understand the following (...
12
votes
2answers
951 views
Why do we care about $(\infty,2)$-categories?
crossposted from MSE as suggested by Igor Sikora
Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable ...
1
vote
0answers
61 views
Regular mapping space vs continuous mapping space for affine schemes
Let $A$ and $B$ be Zariski open affine sub-schemes of $\mathbb{A}_{\mathbb{C}}^{n}$ and $\mathbb{A}_{\mathbb{C}}^m$ respectively. We denote the infinite symmetric product of $B$ by $Sym^{\infty}(B)$. ...
1
vote
0answers
134 views
Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?
In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...
6
votes
1answer
454 views
Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
A commutative monoid with zero is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal ...
6
votes
1answer
175 views
Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces
The 1991 paper of Lewis, “Is there a convenient category of spectra?” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:
There is a symmetric ...
0
votes
1answer
188 views
How to compute singular homologies of affine hypersurface in $A^4$ [closed]
I was trying to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$. Can anyone help me computing that? Can anyone tell me some reference ...
9
votes
1answer
195 views
Group completion of $\mathbb{E}_{\infty}$-monoids via tensor products
$\newcommand{\K}{\mathrm{K}}$The abelian group completion functor $\K_0\colon\mathsf{CMon}\to\mathsf{Ab}$ satisfies
$$
\K_0(A)
\cong
\mathbb{Z}\otimes_{\mathbb{N}}A,
$$
naturally in $A\in\mathrm{Obj}(\...
5
votes
0answers
218 views
Existence of an affine variety with homotopy type of suspension of another affine variety
Let $X$ be an affine variety. My question is does there exist another affine variety with the homotopy type of the suspension of $X$?
9
votes
1answer
336 views
Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...
8
votes
0answers
173 views
Classifying spaces of monoidal categories and deloopings
$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\...
9
votes
1answer
239 views
Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles
I am trying to visualize the genus-two Riemann surface given by the curve
$$
y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}.
$$
We can regard this surface as a three-fold cover of the sphere with four ...
