All Questions
Tagged with homotopy-theory stable-homotopy
261
questions
8
votes
0
answers
391
views
+50
Descent vs effective descent for morphisms of ring spectra
Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...
- 403
25
votes
3
answers
2k
views
Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
- 1,646
7
votes
1
answer
244
views
Proof of the equivalence of spectra $(\mathbb{S}^{-1} \otimes \mathbb{S}^{-1})_{h \Sigma_2} \cong \Sigma^{-1} \mathbb{RP}_{-1}^{\infty}$
$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\Th}{Th}$I am trying to give a hands-on proof of the equivalence of spectra in the title. I am using the definitions $\mathbb{RP}^{\infty}_{-...
- 9,507
8
votes
0
answers
209
views
A few questions about Priddy’s construction of $BP$
In A Cellular Construction of BP and Other Irreducible Spectra, Priddy gives an interesting approach to constructing the Brown-Peterson spectrum $BP$. His result is often summarized as
If you start ...
- 62.6k
5
votes
0
answers
188
views
Identifying a map in a fiber sequence
Let $Q = \Omega^{\infty} \Sigma^{\infty}$ be the stabilization functor. Suppose we have a sequence of maps $Q \mathbb{RP}^{n-1} \to Q \mathbb{RP}^{n} \to QS^n$
and suppose we know that it is a fiber ...
- 223
6
votes
0
answers
316
views
On the nilpotence of the attaching maps for $\mathbb C \mathbb P^\infty$
Consider the usual cell structure on $\mathbb C \mathbb P^\infty$. The skeleta are the $\mathbb C \mathbb P^n$’s, and there is one cell in each even degree. So we have cofiber sequences $S^{2n+1} \to \...
- 62.6k
14
votes
1
answer
344
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.3k
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 21
4
votes
2
answers
376
views
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
7
votes
4
answers
391
views
Why is the first nontrivial $p$-local stable stem cyclic?
Let $\pi_\ast^{(p)}$ be the ring of $p$-local stable homotopy groups of spheres. This is a nonnegatively graded ring, with $\mathbb Z_{(p)}$ in degree $0$. The first nonvanishing positive degree ...
- 3,506
22
votes
3
answers
2k
views
Stable homotopy type theory?
This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
- 5,009
26
votes
1
answer
820
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 ...
- 6,258
3
votes
0
answers
78
views
Shearing maps on domain of assembly map in algebraic $K$-theory
Let $H \to G$ be an inclusion of abelian groups, and let $R$ be a ${\Bbb Z}[H]$-algebra. Assume that the assembly map ${\Bbb S}[BG] \otimes_{\Bbb S} K(R \otimes_{{\Bbb Z}[H]} {\Bbb Z}[G]) \to K((R \...
7
votes
1
answer
378
views
Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
Let $Sp$ be the category of spectra. Let $L : Sp \to Sp_L$ be the localization functor onto a reflective subcategory.
Question 1: Is it ever the case that $L(S^0)$ is not bounded below?
Question 2: ...
- 55.1k
8
votes
2
answers
568
views
Derivations in the Steenrod algebra
Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.
Question 1:
Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?
Question 2: Is there a classification of ...
- 4,186