All Questions
Tagged with homotopy-theory stable-homotopy
261
questions
4
votes
2
answers
376
views
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
5
votes
1
answer
209
views
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...
- 11.5k
4
votes
1
answer
387
views
The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)
Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm ...
- 11.5k
8
votes
1
answer
579
views
Why isn't the anchor map in Lurie's "Rotation Invariance in Algebraic K-Theory" zero?
I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by
$$\...
- 455
8
votes
0
answers
352
views
Worst-case complexity of calculating homotopy groups of spheres
Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...
- 827
5
votes
0
answers
435
views
Is there anything special about the Honda formal group?
The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
- 1,837
5
votes
1
answer
208
views
Rational G-spectrum and geometric fixed points
For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
2
votes
0
answers
406
views
About infinite loop space and $\Omega$ spectrum
Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure.
$BA$ is a classifying space of the topological abelian monoid.
My purpose is to construct an infinite ...
- 121
3
votes
1
answer
165
views
Can the Picard-graded homotopy of a nonzero object be nilpotent?
Let $\mathcal C$ be a symmetric monoidal stable category such that the thick subcategory generated by the unit is all of $\mathcal C$ -- in particular, every object is dualizable (I'm particularly ...
- 62.6k
4
votes
1
answer
205
views
If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory?
$\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A_\ast : \Spt \to \GrAb$ is a strong ...
- 62.6k
6
votes
0
answers
233
views
Flatness of objects in a prestable $\infty$-category
I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions?
The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...
- 2,446
5
votes
0
answers
356
views
Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?
Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....
4
votes
1
answer
201
views
Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category
Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of ...
- 728
8
votes
0
answers
360
views
What is the Balmer spectrum of the p-complete stable homotopy category?
When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
- 1,837
3
votes
0
answers
126
views
Equivariant spectra with coefficients
In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with ...
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