Research Statement
Emanuele Dotto
2015
Overview
My research interests range between algebraic K-theory and equivariant homotopy theory. I
am currently extending the trace methods of [16],[30],[49] to the Real algebraic K-theory of
rings and ring spectra with Wall antistructures of [39], in parallel with an equivariant theory
of Goodwillie calculus of functors [35]. My current main objective is to prove a DundasGoodwillie-McCarthy theorem for Real and Hermitian algebraic K-theory, as well as relating
equivariant calculus to a theory of equivariant operads. The interplay between Real K-theory
and equivariant calculus should lead to calculations in Hermitian K-theory of rings [43] and
in Hermitian A-theory [55], and on the long term it should find important applications in
geometric topology and in surgery theory.
The document is divided into two sections, one on K-theory and one on calculus. Each
section is divided into a background part and a part specific to my research project. The
research goals are divided into short and longer term goals. The short term goals are the main
problems that I am currently investigating, and that I believe they could be solved within one
or two years from now. The longer term goals are bigger problems which arise naturally from
the short term goals.
1
1.1
Real K-theory and traces
Background
Higher algebraic K-theory was introduced in the seminal work of [51]. It is a construction that
associates to a ring A a space (or spectrum) K(A) which is a topological group completion for a
moduli space of finitely generated projective modules over A. This theory was later extended in
[56] to ring spectra, and in fact to a more general input known today as Waldhausen categories.
The homotopy groups of the K-theory of a Waldhausen category appear in various fields of
mathematics: in algebraic geometry and in number theory where the input is the category
of perfect modules over a scheme or the ring of integers in a number field ([54] and [59]), in
algebra where the input is a group-ring ([4]), and in geometric topology where the input is the
category of finite retractive CW -complexes over a space or the spherical group-ring of a loop
space ([56], [57] and [32]). In each of these cases algebraic K-theory is related to important
open problems of arithmetic and geometric nature.
Not surprisingly, such an interesting invariant turns out to be extremely difficult to calculate. Even the algebraic K-theory of the ring of integers is not quite completely understood
(it is up to the Kummer-Vandiver conjecture). A successful strategy for computations is to
compare algebraic K-theory to objects of a more homotopy-theoretical nature where the computations are somewhat more approachable. These objects are typically Bökstedt’s topological
Hochschild homology THH of [17] or the topological cyclic homology TC of [16]. One successful comparison tool is the Dundas-Goodwillie-McCarthy Theorem of [49], [28] and [29] which
compares K-theory and topological cyclic homology via the trace map Tr : K → TC of [16].
This theorem states that for suitable maps of ring spectra A → B the corresponding square
1
Research Statement
Emanuele Dotto
2015
relating K-theory and TC
K(A)
Tr
K(B)
Tr
/ TC(A)
/ TC(B)
is homotopy cartesian after completion at a prime. These “trace methods” led to several
computations, for example of the K-theory of perfect fields in [38], of truncated polynomial
algebras in [37] and [1], or of the sphere spectrum in [44],[52] and [13].
Hermitian K-theory is a similar invariant, where the input ring is replaced by a ring
with a Wall antistructure [58] (e.g. a commutative ring). It was first defined in [43] as a
topological group-completion of a moduli space of Hermitian forms. Similar to algebraic Ktheory, Hermitian K-theory has great arithmetic and geometric relevance. The Hermitian
K-theory of an Abelian number field is related to some special values of the Dedekind zetafunction ([9]), and the Hermitian K-theory of an integral group-ring is related (up to quadratic
refinement) to the surgery exact sequence. Karoubi’s definition was later extended to exact
categories with duality ([41] and [53]) and to a Hermitian K-theory of spaces in [55], again
with important relations to geometric topology and surgery ([60],[61] and [62]).
Also in the Hermitian case the calculations are hardly accessible, and a theory of traces is
still lacking.
1.2
The research project
The starting point of my research is Hesselholt and Madsen’s recently developed theory of Real
K-theory [39]. From an exact category with weak equivalences C and duality D : C op → C ,
they construct a Z/2-equivariant spectrum KR(C ), whose underlying spectrum is the K-theory
K(C ) and whose fixed points spectrum is the Hermitian K-theory of C
KR(C )Z/2 ≃ KH(C )
This improves earlier constructions of [53] and [62] where a similar Real K-theory was constructed as a Z/2-space. Having a theory valued in stable Z/2-equivariant homotopy theory
is a main prerequisite for a theory of traces.
In my PhD thesis [20] I developed a Z/2-equivariant analogue of THH for spectral categories
with duality: the Real topological Hochschild homology THR. Its Z/2-fixed points spectrum
plays the role of a Hermitian version of THH. I showed in [20] that THR enjoys structural
properties which are analogous to the classical theory of [31] and, for linear categories, I
constructed a Z/2-equivariant trace map KR → THR. In particular this provides a natural
map from Hermitian K-theory
KH(A) −→ THR(A)Z/2
for every (discrete) ring with Wall-antistructure A. The spectrum THR has a natural S 1,1 ⋉
Z/2-action, where S 1,1 is the sign-representation sphere. The Real topological cyclic homology
TCR is a Z/2-spectrum defined from the action on THR of the finite subgroups of S 1 .
2
Research Statement
Emanuele Dotto
2015
Short term goal 1. Find a convenient collection of maps of rings with antistructure f : A → B
for which the square of Z/2-spectra
KR(A)
f∗
Tr
KR(B)
/ TCR(A)
Tr
f∗
/ TCR(B)
is homotopy cartesian after suitable completion.
Progress. I am following the strategy of [49], of proving the result for “split square-zero extensions” A ⋉ M → A first, and then enlarging the class of maps by an algebraic inductive
argument on the nilpotence of the kernel.
In the split square-zero extensions case, the main result of my PhD thesis [20] shows that
the trace map defines an equivalence between a suitable stabilization of KR and THR. This is
the first place where the equivariant calculus of functors of §2 intertwines with Real K-theory,
as this stabilization arises as a Z/2-equivariant derivative of the functor K(A ⋉ M (−)). This
is analogous to the main result of [30], which constitutes the first step of the proof of [49].
The next step is to show that THR is also the Z/2-equivariant derivative of TCR. This
should use methods which are similar to [36], and partial results are already included in my
thesis. This would prove that the square in question is homotopy cartesian after taking Z/2derivatives. The final step is to use equivariant calculus of functors to remove the derivatives. I
developed a calculus machinery sufficient for this procedure in [21], together with the necessary
analytical properties of Real K-theory. The corresponding analytical properties of TCR still
need to be explored.
The Real K-theory of [39] is defined for exact categories with duality. A similar theory
had already been established in [62] for a larger class of Waldhausen categories “with SpanierWhitehead duality”. This theory was constructed to model the Hermitian K-theory of spaces
of [55]. The construction of [62] solves some very technical issues about the non-functoriality
of the Spanier-Whitehead dual of a space, but it is rather indirect and it does not seem to
produce a Z/2-equivariant spectrum.
Short term goal 2. Extend the construction of Real K-theory to a convenient class of “Waldhausen categories with duality”. This class of categories should include the category of finite
modules over a ring spectrum A with the function spectrum “duality” FA (−, A), and support a
trace map KR(A) → TCR(A) of Z/2-equivariant spectra.
Progress. I am combining the constructions of [62] and [39]. The central idea is to define a
Waldhausen structure on the strictification construction “xC” of [62]. There are many technical
issues related to the existence of structures (pullbacks, duality,...) only “up to equivalence”.
When set it up correctly, these issues should be taken care of by a mix of the S•2,1 -construction
of [39] and Blumberg and Mandell’s homotopical S• -construction of [14].
Longer term goal 3. Study involutions on Witt vectors and generalize the computations of
[38] of the K-theory of the dual numbers of perfect fields to the Real context.
3
Research Statement
Emanuele Dotto
2015
Strategy. The strategy to follow would be the one of [38], using the Real Dundas-McCarthy
theorem above. It involves a calculation of the homotopy groups of the Z/2-fixed points of
TCR(k[ϵ]/ϵ2 ) in terms of the ring of Witt vectors W (k). An intermediate stage is the fixedpoints analogue of the isomorphism of [38] between π∗ THH(k)Cpn and the symmetric algebra
SWn+1 (k) {σ}, where Wn+1 (k) is the ring of Witt-vectors of length n + 1 and σ is a generator
of degree 2.
Longer term goal 4. Extend the Real Dundas-Goodwillie-McCarthy Theorem to ring spectra
with antistructures.
Strategy. This would either involve an approximation of ring spectra with antistructure as
limits of simplicial rings with antistructure similar to [28], or a “spectral proof” of the DundasGoodwillie-McCarthy theorem for suitable radical extensions of ring spectra with antistructures. The second approach involves an version of the theorem for the K-theory of exact
infinity-categories, which is work in progress in a joint project with Clark Barwick [6].
An immediate consequence of the extension of Real K-theory to ring spectra is the existence
of a Z/2-spectrum KR(SZ/2 ∧ G+ ), where SZ/2 is the Z/2-equivariant sphere spectrum and G
is a topological group with the anti-involution defined by inversion. When G = ΩX is a loop
space, the underlying spectrum is equivalent to the A-theory spectrum A(X), and the Z/2fixed-points should be equivalent to Vogell’s Hermitian A-theory AH(X) of [55]. Classically
A(X) splits as
A(X) ≃ (S ∧ X+ ) ∨ W h(X)
where W h(X) is the Whitehead spectrum. When X = M is a manifold W h(M ) is a geometric
object related to a stabilization of the space of h-cobordisms of M ([57]). Moreover the splitting
is realized through the trace map K(S ∧ G+ ) → THH(S ∧ G+ ). The spectrum W h(M ) has
a natural involution whose homotopy orbits are related to the diffeomorphism group of M ,
through surgery theory ([60] and [61]). Thus an equivariant understanding of the splitting of
A(M ) could lead to calculations of the diffeomorphism group of M .
Longer term goal 5. See if the Real trace map extends the splitting A(X) ≃ (S∧X+ )∨W h(X)
to an equivariant splitting
KR(SZ/2 ∧ ΩX+ ) ≃ (SZ/2 ∧ X+ ) ∨ RW h(X)
where RW h(X) is the Whitehead spectrum with the involution of [60] and [61]. On fixed-points,
this would induce a splitting of Vogell’s Hermitian A-theory AH(X).
Strategy. This is an ambitious goal. The proof of [57] is mostly categorical, and so is the
construction of the involution on KR described in the short term Goal 2, giving a chance to
the arguments of [57] to be extended to the Real context.
Longer term goal 6. Use the Real trace and the assembly map in THR to obtain information
about the Hermitian assembly map KH(Z) ∧ BG+ → KH(Z[G]).
4
Research Statement
Emanuele Dotto
2015
Strategy. A successful strategy for the study of the K-theoretic assembly map is to study the
assembly of spherical group-rings K(S) ∧ BG+ → K(S[G]), as in [15]. Similarly, the Real Ktheory assembly KR(SZ/2 )∧BG+ → KR(S[G]) and its THR counterpart could retain important
information about the integral Hermitian assembly map, hopefully rational injectivity. These
kind of results are highly relevant in geometric topology [45]. This is work in progress with
Crichton Ogle [26].
2
2.1
Equivariant calculus of functors
Background
Calculus of functors was first introduced by Goodwillie in [33] and [34] with a primary focus
on algebraic K-theory of spaces. It was later refined with a broader perspective in [35] by
the same author, and it was extended to a more abstract homotopy theoretical framework in
[10] and [11]. The main idea of this theory is to construct, from a homotopy invariant functor
F : C → D between two homotopy theories C and D (typically pointed spaces or spectra), a
tower of functors
. . . −→ Pn+1 F −→ Pn F −→ . . . −→ P1 F
that mimics the Taylor expansion of a real-valued function. The functor Pn F : C → D is “the
best approximation of F by an n-homology theory”, and it is called the n-excisive approximation of F . The n-excision property is defined in terms of the behavior of a functor on certain
cubical diagrams in C of dimension n + 1. In particular P1 F is a homology theory, in the sense
that it sends homotopy pushout squares in C to homotopy pullback squares in D.
There are two fundamental theorems that are proved in [35]. The first finds a technical
condition on a functor on pointed spaces F : Top∗ → Top∗ that insures that the tower above
converges to F
≃
F (X) −→ holim Pn F (X)
n
for spaces X which are sufficiently highly-connected. Examples of functors that satisfy this
condition include the identity functor, the algebraic K-theory of spaces functor A(−), the
e ⋉ M (−)) from Goal 1, and the stable mapping space
relative algebraic K-theory functor K(A
∞
∞
functor Ω Σ Map∗ (K, −) out of a finite pointed CW -complex K. The second fundamental
result is the characterization of the layers Dn F = hofib(Pn F → Pn−1 F ) of the tower above,
as functors of the form
Dn F (X) ≃ (CF ∧ X ∧n )hΣn
for some spectrum CF with a naïve Σn -action. Here Σn acts on the smash powers of X by
permuting the factors, and (−)hΣn denotes the homotopy orbits.
This theory has a big impact in algebraic K-theory and in unstable homotopy theory.
It is applied in [49] to the algebraic K-theory and to the topological cyclic homology functors to prove the Dundas-Goodwillie-McCarthy Theorem from §1.1. The trace map in Atheory A(X) → THH(S ∧ ΩX+ ) is equivalent to the first excisive approximation map A(X) →
P1 A(X), and it is used in [56] to construct the A-theory splitting of §1.2. The layers of the
5
Research Statement
Emanuele Dotto
2015
tower of the identity functors are described in [42], and they are used in [3] to calculate the
periodic homotopy of odd-dimensional spheres.
Now suppose that G is a finite group an that F : C G → D G is a functor between Gequivariant homotopy theories (typically the categories of pointed G-spaces or of genuine Gspectra). Standard Goodwillie calculus can certainly be applied to F , but not without disapG
pointment. For example the layer Dn F of a homotopy invariant functor F : TopG
∗ → Top∗
G
is classified by a spectrum in the category Top∗ (with a Σn -action), that is to say a naïve
G-spectrum. What we would like is a theory where the layers of the tower are genuine Gspectra with some sort of Σn -action. In particular if F sends the point to the point there is
an equivalence
P1 F (X) ≃ hocolim Ωn F (Σn X)
n
but a “genuine theory” would require a stabilization of the form hocolimV ΩV F (ΣV X) where
V runs through the finite dimensional representations of G.
2.2
The research project
In spent the last couple of years researching on this subject, resulting in four papers [21], [22],
[25] and [23]. The first paper studies the interaction between equivariant calculus and Real
algebraic K-theory of rings that is needed for the Real Dundas-Goodwillie-McCarthy Theorem
of Goal 1. The other papers develop a theory of equivariant calculus based on G-indexed cubes.
The central idea for developing this theory is to replace the (n + 1)-cubes used in [35]
to define n-excision with cubes which are indexed on finite G-sets. This requires a theory
of “genuine” equivariant homotopy limits and colimits for diagrams which are indexed on a
category with a G-action, and the very notion of an “equivariant model category”. These
foundations have been laid out in [25] in a joint project with Kristian Moi. In [23] I develop
a notion of higher order equivariant excision for functors F : C G → D G between equivariant
homotopy theories, and an equivariant version of the Goodwillie tower which is indexed on
finite G-sets. In particular by considering only the free G-sets this gives a tower
. . . −→ P(n+1)G F −→ PnG F −→ . . . −→ PG F
where PG F is a genuine equivariant homology theory. Among other results the paper contains
a calculation of the layers of this tower for the identity functor on pointed G-spaces in terms of
the partition complexes, analogous to [42] and [3]. There are many angles and developments
of the theory that still need to be investigated.
g ⋉
Short term goal 7. Calculate the equivariant tower of the Real K-theory functor KR(A
Z/2
G
M (−)) : Top∗ → SpZ/2 and of the equivariant A-theory functor AG : TopG
∗ → Sp of [50].
g ⋉ M (−)) of the
Progress. I proved in my thesis and in [21] that the first stage PZ/2 KR(A
equivariant tower of the Real K-theory of a split square zero extension A⋉M → A is equivalent
to the Real topological Hochschild homology functor THR(A; M (S 1,1 ∧(−))). The higher stages
should be described in terms of a Real TR-theory with coefficients, analogous to the work of
[48].
6
Research Statement
Emanuele Dotto
2015
The equivariant A-theory AG of [50] is a functor whose fixed-points AG (X)H by a subgroup
H of G is the K-theory of the category of finite retractive H-spaces over X. In a joint project
with Mona Merling [24] we are trying to describe the equivariant deloopings of AG in terms
of iterations of the S• -construction indexed on G-sets. This would give a better description
of AG , and it is a starting point for understanding the layers of the equivariant tower of AG .
These should be described in terms of equivariant loop spaces of the smash powers of X.
G
Short term goal 8. Classify “J-homogeneous” functors TopG
∗ → Top∗ for a finite G-set J.
Progress. The first step in Goodwillie’s classification is to show that n-homogeneous functors
take value in infinite loop spaces. I proved the equivariant analogue of this result in [23]:
“J-homogeneous” functors deloop by the permutation representation of J. Let us suppose for
simplicity that J = nG is free. Genuine G-spectra with a naïve Σn -action classify functors that
are nG-excisive and (n − 1)-reduced [23], whereas nG-homogeneous functors are by definition
nG-excisive and (n − 1)G-reduced. The calculation of the layers of the equivariant tower of
the identity of [23] suggests that nG-homogeneous functors should be of the form
Ω∞G ((C ∧ X ∧n ) ∧Σn EFn )
where Ω∞G is the genuine infinite loop space functor, and EFn is a classifying space for the
family Fn of the subgroups of G×Σn whose quotients are Σn -free. Here C is an “Fn -spectrum”,
a G × Σn -spectrum whose homotopy type is determined by the fixed-points spectra for the
subgroups that belong to Fn . In the language of [5] this is a Mackey functor on the full
subcategory of the orbit category of G spanned by Fn . It is still unclear what kind of family
of subgroups should appear for a general G-set J. This classification is work in progress in a
joint project with Tomer Schlank [27].
Longer term goal 9. Find a suitable notion of “equivariant analytic functor” that guarantees
G
the convergence of the equivariant Goodwillie tower of a functor F : TopG
∗ → Top∗ .
Progress. I proved in [23] that this convergence issue can be reduced to Goodwillie’s notion
of analytic functors if F “commutes with fixed-points”. This is for example the case for the
identity functor, but not for the functors of Goal 7. This condition should be phrased in terms
of equivariant connectivity estimates similar to the equivariant Blakers-Massey Theorem of
[22].
Another important feature of Goodwillie calculus is the relationship with operads. The
derivatives of the identity functor on pointed spaces have the structure of an operad in spectra,
which is dual to the co-operad structure on the partition complexes, and whose homology is
the Lie operad [18]. Moreover the chain rule in functor calculus equips the derivatives of a
homotopy functor with the structure of a module over this operad [2]. The calculation of the
equivariant derivatives of the identity on pointed G-spaces of [23] together with the conjectured
classification of homogeneous functors of Goal 8 suggests that a similar chain rule should exist
in the equivariant setting.
Short term goal 10. Develop a homotopy theory of “genuine” equivariant operads in an
equivariant model category, and of algebras and modules over them.
7
Research Statement
Emanuele Dotto
2015
Strategy. This is an ongoing project joint with Tomer Schlank [27]. Classically a G-operad, say
in spaces, is simply defined as an operad in the category of G-spaces ([47] and [19]). Although
these are the right objects, the levelwise equivalences of G-spaces produce a homotopy theory
of “naïve” G-operads. The classification of Goal 8 suggests that an equivalence of G-operads
should be defined by the fixed-points for the subgroups of G × Σn that belong to the families
Fn . In this model structure the N∞ -operads of [12], used to encode the multiplicative norms
of G-commutative ring spectra of [40], will be a “Σ-cofibrant” replacement of the commutative
operad. We are constructing these model structures on operads, algebras and modules by
generalizing the transport techniques of [8] and [46].
Additionally, [7] contains a definition of G-operad (in the infinity categorical setting) in
terms of multi-operations which are indexed on finite G-sets. We are in the process of proving
that the two approaches are in fact equivalent.
The calculation of the equivariant derivatives of the identity of [23] shows that the equivariant derivatives of the identity functor of pointed G-spaces have the structure of a G-operad,
induced by the co-operad structure on the partition complexes of [18].
Longer term goal 11. Prove a chain rule for the equivariant derivatives of homotopy functors,
and relate the homology of the derivatives of the identity to an equivariant Lie operad.
Strategy. In joint work with Tomer Schlank [27]. The equivariant derivatives of a composition
of functors should be equivalent to the composition product of symmetric sequences as in [2].
By “equivalent” we now mean in the sense of Goal 10, with respect to the subgroups of G × Σn
which belong to the families Fn . It is still unclear if the homology of the derivatives of the
identity correspond to a known algebraic structure of G-Lie operad.
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