Note: The papers here are more likely to be up to date than the versions on the
On categories of slices
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
as a module over the endomorphism Green functor
; (ii) the category of n-slices is equivalent to the full
subcategory of right modules over [W,W]
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
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.
Power operations for HF2 and
a cellular construction of BPR.
Submitted for publication.
We develop a bit of the theory of power operations
-equivariant homology with constant coefficients at
In particular, we construct RO(C2
)-graded Dyer-Lashof operations
and study their action on an equivariant dual Steenrod algebra.
As an application, we give a cellular construction of BPR
We study some power operations for ordinary C2
homology with coefficients in the constant Mackey functor at F2
. In addition to a few foundational
results, we calculate the action of these power operations on a C2
-equivariant dual Steenrod
algebra. As an application, we give a cellular construction of the C2
its slice tower.
Orientations and Topological Modular Forms with Level Structure
Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from
-genera valued in topological modular forms with level structure. We give
examples of such series for tmf0
(N) and show that the Ochanine genus comes from an
-ring map. We also show that, away from 6, certain tmf orientations of
MString descend to orientations of MSpin.