Examples of disk algebras. Feb 2023. With
Sanath Devalapurkar, Jeremy Hahn, Tyler Lawson,
and Andrew Senger.
We produce refinements of the known multiplicative structures on the
Brown--Peterson spectrum BP, its truncated variants BP<n>,
Ravenel's spectra X(n), and evenly graded polynomial rings over the sphere
spectrum. Consequently, topological Hochschild homology relative to these rings
inherits a circle action.
For a prime number p and a p-quasisyntomic commutative ring R, Bhatt--Morrow--Scholze defined motivic filtrations on the p-completions of THH(R),TC-(R),TP(R), and TC(R), with the associated graded objects for TP(R) and TC(R) recovering the prismatic and syntomic cohomology of R, respectively. We give an alternate construction of these filtrations that applies also when R is a well-behaved commutative ring spectrum; for example, we can take R to be S, MU, ku, ko, or tmf. We compute the mod (p,v1) syntomic cohomology of the Adams summand and observe that, when p=3, that the corresponding motivic spectral sequence collapses at the E2-page.
We equip BP<n> with an E_{3}-BP-algebra structure, for each prime p and height n. The algebraic K-theory of this E_{3}-ring is of chromatic height exactly n+1.
Specifically, it is an fp-spectrum of fp-type n+1, which can be viewed as a higher height version of the
Lichtenbaum-Quillen conjecture.
We compute the C_{p}-equivariant dual Steenrod algebras associated to the constant Mackey functors
F_{p} and Z_{(p)}, as Z_{(p)}-modules. The C_{p} mod p dual Steenrod algebra
is not a direct sum of RO(C_{p})-graded suspensions of F_{p} when p is odd, in contrast with the classical
and C_{2}-equivariant dual Steenrod algebras.
We define, in C_{p}-equivariant homotopy theory for p>2, a notion of µ_{p}-orientation analogous to a C_{2}-equivariant Real orientation. The definition hinges on a C_{p}-space CP_{µp}, which we prove to be homologically even in a sense generalizing recent C_{2}-equivariant work on conjugation spaces.
We prove that the height p-1 Morava E-theory is µ_{p}-oriented and that tmf(2) is µ_{3}-oriented. We explain how a single equivariant map S^{2ρ} → Σ^{∞}CP_{µp} completely generates the homotopy of E_{p-1} and tmf(2), expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.
We give a new proof, independent of Lin’s theorem,
of the Segal conjecture for the cyclic group of order two.
The key input is a calculation, as a Hopf algebroid, of
the Real topological Hochschild homology of F_2.
This determines the E2-page of the descent spectral sequence for
the map from NF_2 to F_2, where NF_2 is the C2-equivariant Hill–Hopkins–Ravenel norm
of F_2.
The E2-page represents a new upper bound on the RO(C_2)-graded homotopy of NF_2,
from which the Segal conjecture is an immediate corollary.
In this mostly expository note we take advantage
of homotopical and algebraic advances
to give a modern
account of power operations on the mod 2 homology
of commutative ring spectra. The main
advance is a quick proof of the Adem
relations utilizing the Tate-valued Frobenius as a homotopical
incarnation of the total power operation. We also give a
streamlined derivation of the action of power operations
on the dual Steenrod algebra.
We prove that the G-equivariant mod p Eilenberg--MacLane
spectrum arises as an equivariant Thom spectrum for any finite,
p-power cyclic group G, generalizing a result of Behrens and
the second author in the case of the group C_{2}.
We also establish a construction of HZ_{(p)},
and prove intermediate results that may be of independent interest.
Highlights include an interesting action on quaternionic
projective space, and an analysis of the extent to which the
non-equivariant HF_{p} arises as the Thom spectrum
of a more than double loop map.
We prove that the C_{2}-equivariant Eilenberg-MacLane
spectrum associated with the constant Mackey functor
F_{2} is equivalent to
a Thom spectrum over Ω^{ρ}S^{ρ+1}.
In this note we state corrected and expanded versions of
our previous results on
power operations for C_{2}-equivariant Bredon homology
with coefficients in the constant Mackey functor with mod 2 coefficients.
In particular, we give a version of the Adem relations. The proofs
rely on certain results in
equivariant higher algebra which we will supply in a longer
version of this paper.
We prove that if R is an E_{2}-ring with homotopy
concentrated in even degrees, and {x_{j}} is a sequence
of elements in even degrees, then R/(x_{1}, ...) admits
the structure of an E_{1}-R-algebra. This
removes an assumption, common in the literature, that
{x_{j}} be a regular sequence.
In this paper we give an algebraic description
of the category of n-slices for an arbitrary
group G, in the sense of Hill-Hopkins-Ravenel.
Specifically, given a finite group G and an integer n, we construct
an explicit G-spectrum
W (called an isotropic slice n-sphere)
with the following properties: (i) the n-slice of a G-spectrum X
is equivalent to the data of a certain quotient of the Mackey functor
[W, X] as a module over the endomorphism Green functor
[W,W]; (ii) the category of n-slices is equivalent to the full
subcategory of right modules over [W,W] for which
a certain restriction map is injective. We use this theorem to recover
the known results on categories of slices to date, and exhibit
the utility of our description in several new examples. We go
further and show that the Green
functors [W,W] for
certain slice n-spheres have a special property
(they are geometrically split)
which reduces the amount of data necessary
to specify a [W,W]-module. This step
is purely algebraic and may be of independent interest.
We develop a bit of the theory of power operations
for C_{2}-equivariant homology with constant coefficients at
F_{2}.
In particular, we construct RO(C_{2})-graded Dyer-Lashof operations
and study their action on an equivariant dual Steenrod algebra.
As an application, we give a cellular construction of BPR,
after Priddy.
We study some power operations for ordinary C_{2}-equivariant
homology with coefficients in the constant Mackey functor at F_{2}. In addition to a few foundational
results, we calculate the action of these power operations on a C_{2}-equivariant dual Steenrod
algebra. As an application, we give a cellular construction of the C_{2}-spectrum BPR
and deduce
its slice tower.
Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from
E_{∞}-genera valued in topological modular forms with level structure. We give
examples of such series for tmf_{0}(N) and show that the Ochanine genus comes from an
E_{∞}-ring map. We also show that, away from 6, certain tmf orientations of
MString descend to orientations of MSpin.