All Questions
Tagged with homotopy-theory stable-homotopy
260
questions
7
votes
1
answer
236
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}_{-...
- 223
8
votes
0
answers
206
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
186
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
298
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
7
votes
4
answers
389
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 ...
- 62.6k
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
372
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: ...
- 62.6k
8
votes
2
answers
565
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 ...
- 62.6k
2
votes
0
answers
166
views
Infinite loop space as an endofunctor of compactly generated weak hausdorff topological spaces?
I am trying to see whether it is possible to define smash product of infinite loop spaces using the space $S^{\infty}$.
Let C be the category of compactly generated weak Hausdorff topological spaces. ...
user30211
3
votes
1
answer
145
views
Can a phantom map have finite cofiber?
Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum?
Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
- 62.6k
1
vote
0
answers
210
views
Properties of colim Ωⁿ Σⁿ X
I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of the endofunctor $Q: ...
user30211
1
vote
1
answer
226
views
Symmetric-monoidal-associative smash product up to homotopy
I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.
Recall that a sequential ...
user30211
15
votes
1
answer
768
views
If homotopy groups of spaces are identical, then stable ones are also identical?
Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this ...
- 6,684
7
votes
0
answers
154
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 ...
- 6,093
6
votes
1
answer
243
views
Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?
It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
- 62.6k