HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY
OF THE CIRCLE
HÅKON SCHAD BERGSAKER
Introduction
Algebraic K-theory of a ring, fully defined by Quillen in his 1973 paper [Qui73], has become one
of the most important invariants of rings, and much effort has gone into computing it. Motivated
by his study of manifolds, Waldhausen some years later defined his A-theory of spaces [Wal78],
and also provided a general framework for algebraic K-theory which included both his A-theory
and Quillen’s K-theory as special cases [Wal85]. The introduction of well-behaved categories of
structured ring spectra in the 90s allowed for a definition of algebraic K-theory of a ring spectrum
(suitably interpreted). This construction recovers Quillen’s K-theory as K(R) = K(HR), where
HR is the Eilenberg-Mac Lane spectrum of the ring R. Waldhausen’s A-theory of a space X is
recovered as A(X) = K(S[ΩX]), where S[ΩX] is the spherical group ring of the loop space of X.
Algebraic K-theory is notoriously difficult to compute, even for ordinary rings, but several fruitful
strategies for its computation has been developed. One such is the so-called trace methods, via
topological cyclic homology and topological Hochschild homology.
Topological Hochschild homolgy of ring spectra was defined by Bökstedt in [Bök87]. It is a
topological version of Hochschild homology for rings, in the sense that its construction is essentially to replace tensor products over the integers with smash products over the sphere spectrum. Originally defined for certain “functors with smash products”, Bökstedt’s construction
works equally well for the more modern concept of a symmetric ring spectrum. There is a trace
map tr : K(B) → T HH(B) from algebraic K-theory to topological Hochschild homology which
can be used to detect elements in π∗ K(B), but it is typically very far from being an equivalence.
Topological cyclic homology was defined by Bökstedt, Hsiang and Madsen [BHM93]. It originally
takes as input a space, or more generally a functor with smash product, but again the construction
also works for more modern categories of structured ring spectra. Topological cyclic homology is a
refinement of topological Hochschild homology in the sense that there is a map T C(B) → T HH(B)
and the trace map tr : K(B) → T HH(B) lifts to a cyclotomic trace map trc : K(B) → T C(B).
It turns out that the map trc can detect a lot of K(B), and after a suitable completion it can
even be an equivalence on connective covers. (T C is in general only (−2)-connected.) This is very
useful since T C, although not easy to compute, is approachable by standard homotopy theoretical
methods, such as spectral sequences.
On the geometric side, Waldhausen’s A-theory of spaces is related to automorphisms of manifolds. Write A(X) for the spectrum-valued A-theory of a space X, and W hCAT (M ) for the
Whitehead spectrum of a CAT -manifold M . Here CAT is either DIF F , T OP or P L, denoting
the category of smooth, topological or piecewise linear manifolds, respectively. Results by Burghelea and Lashof [BL74] and Kirby and Siebenmann [KS77] imply that W hT OP (M ) ' W hP L (M ),
and there is a cofiber sequence
(1)
A(∗) ∧ M+ → A(M ) → W hP L (M ) .
In the smooth case there is a splitting
DIF F
A(M ) ' Σ∞
(M ) ,
+ M ∨ Wh
Date: April 18, 2016.
1
2
HÅKON SCHAD BERGSAKER
which in the case M = ∗ serves to bootstrap the T OP and P L case. (The case M = ∗ for T OP
and P L is trivial since W hP L (∗) ' ∗ by the so-called Alexander trick.) The underlying infinite
loop space of the spectrum W hCAT (M ) is constructed as the double delooping of C CAT (M ), the
space of stable concordances (also known as pseudo-isotopies) of M . By definition, C CAT (M ) =
hocolimn C CAT (M × I n ), where the concordance space C CAT (M ) is the space of automorphisms
of M × I fixing M × 0. Unstably, the T OP and P L case are different in low dimensions, but the
natural map C P L (M ) → C T OP (M ) becomes a homotopy equivalence as soon as dim M ≥ 5.
Since the Whitehead spectra W h(M )CAT encode information about automorphisms of manifolds, they are of great interest in manifold theory. By the link to algebraic K-theory described
above, then, it is important to understand the spectra A(M ), which then can be approached by
studying T C(M ). Particularly important examples are M = ∗ and M = S 1 . This is illustrated
by the result of Farrell and Jones [FJ91] which says that the Whitehead spectra W hT OP (M )
are determined by W hT OP (S 1 ) when M has non-positive sectional curvature everywhere. The
cofiber sequence (1) then motivates the study of the spectra A(∗) and A(S 1 ), and hence T C(∗)
and T C(S 1 ). In particular, it is important to understand their homology, with the various extra
structure it comes with. Since ∗ and S 1 are in fact Lie groups, the spectra T C(∗) and T C(S 1 ) have
naturally defined E∞ ring spectrum structures, which induce Pontrjagin products and Dyer-Lashof
operations in homology. These are the main additional structures we want to study.
In [BR10] we found explicit formulas for the action of the Dyer-Lashof operations (also called
Araki-Kudo operations at the prime 2) on H∗ (T C(∗); F2 ), in addition to describing the Pontrjagin
product and dual Steenrod operations. According to the previous paragraph, the next important
case concerning symmetries of manifolds is the circle. The main goal of this paper is to determine
the homology of T C(S 1 ), including the Pontrjagin product and the actions of the Steenrod and
Dyer-Lashof algebras. Since our proof of this result depends on the corresponding structure for
T C(∗), we provide an argument for describing the multiplicative structure of H∗ (T C(∗); Fp ) at odd
primes. Our proof is modeled on an argument by Mann and Miller in [MM85], where they prove
similar formulas for the homology of the underlying E∞ ring space of the fixed point spectrum
SCp .
Sections 1 and 2 are of a preliminary nature and recall the most important facts about homology
operations and topological cyclic homology, respectively. In Section 3 we give a proof of the
following result about H∗ (T C(∗); Fp ) for odd p, which complements our previous result at the
prime 2.
Theorem 0.1. Let p be an odd prime.
(1) The homology groups of T C(∗) are given as
H∗ (T C(∗); Fp ) = Fp {1, Σβj | j ≥ −1} ,
with |Σβj | = 2j + 1. The Pontrjagin product is trivial, with 1 being the unit.
(2) The dual Steenrod operations in H∗ (T C(∗); Fp ) are given by
j − k(p − 1)
k
P∗ (Σβj ) =
Σβj−k(p−1) .
k
The remaining operations are trivial.
(3) The Dyer-Lashof operations are given by
i
i+j+1 i − 1
Q (Σβj ) = (−1)
Σβi(p−1)+j , for i > j ≥ 0
j
Q0 (Σβ−1 ) = Σβ−1 .
The remaining operations are trivial.
In Section 4 we prove our main result about the multiplicative structure of H∗ (T C(S 1 )). We
do this by comparing with the second stage in the tower defining T C(S 1 ), and also by using the
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
3
above result for T C(∗). We first identify the p-adic homotopy type of T C(S 1 ). The assembly
map mentioned in the following result is a certain multiplicative map T C(A) ∧ BG+ → T C(A[G])
defined for all symmetric ring spectra A and group-like monoids G.
Proposition 0.2. There is an equivalence of spectra
1
T C(S ) ' T C(∗) ∧
1
S+
∨
_
p-n
S
−1
∞
_
!
ΣΣ∞
+ BCpk
,
k=0
1
after p-completion. The inclusion of the summand T C(∗) ∧ S+
coincides with the assembly map.
Theorem 0.3. Let p be any prime.
(1) The mod p homology of T C(S 1 ) is
H∗ (T C(S 1 ); Fp ) = Fp {a−1 (m), 1, a0 , e1 , as (n) | m = 0 or p - m, s ≥ 1, n ∈ Z} ,
with subscripts indicating degree. The Pontrjagin product is trivial, with 1 being the unit.
(2) The dual Steenrod operations in H∗ (T C(S 1 ); Fp ) are given by
s−k
Sq∗k (as (n)) =
as−k (n)
k
for p = 2, and
[s/2] − k(p − 1)
aj−2k(p−1) (n)
k
β∗ (a2j (n)) = a2j−1 (n) if p - n
P∗k (as (n)) =
for odd p. The remaining operations are trivial.
(3) The Dyer-Lashof operations in H∗ (T C(S 1 ); Fp ) are given by
i
i+s+1 i − 1
Q (as (n)) = (−1)
as+i(p−1) (pn)
s
Q0 (a−1 (0)) = a−1 (0) .
The remaining operations are trivial.
A natural question which arises is to what extent the various structures on the homology of
T C(∗) and T C(S 1 ) lift to the homology of the corresponding A-theory spectra, via the cyclotomic
trace map. Unfortunately, in these cases, the induced map in homology of the trace map is almost
trivial. However, in addition to being interesting in itself, the calculations in this paper can be
used to provide key inputs to the calculation of multiplicative structures on the topological cyclic
homology of other ring spectra, via maps S[Z] → B or the unit map S → B. In particular, they
provide crucial information about the multiplicative structures in the homology of T C(Z) and
T C(Z[x±1 ]). This will be addressed in a future paper.
Notation and conventions. Given a set X, we write Fp X for the Fp -vector space with a basis
given by X. We write H∗ (−) for mod p homology H∗ (−; Fp ), and assume all spaces and spectra
are p-complete, except possibly in Section 1. Where necessary, p-completion is implicitly applied.
The circle group of complex numbers of modulus 1 is denoted T .
1. E∞ ring spectra and homology operations
Unless otherwise noted, spectra will mean Lewis-May spectra and E∞ ring spectra will mean
spectra with an action of an E∞ operad. We will use these notions both equviariantly and nonequivariantly. For more details, see [LMSM86]. Exceptions are the K-theory construction used
below, which outputs a symmetric spectrum, and topological Hochschild homology, which takes a
symmetric ring spectrum as input. For more about symmetric spectra, see [HSS00].
4
HÅKON SCHAD BERGSAKER
We use S to denote the sphere spectrum in Lewis-May or symmetric spectra, depending on the
context, and we write S[G] for the unreduced suspension spectrum Σ∞
+ G when G is a topological
monoid, or more generally an A∞ space. The Lewis-May spectrum S[G] is then an A∞ ring
spectrum, which is E∞ if G is a commutative monoid or merely an E∞ space. This construction
also makes sense for symmetric spectra, provided G is a strict (possibly commutative) monoid.
We briefly recall how the mod p homology of an E∞ ring spectrum is augmented by Dyer-Lashof
operations. Let O be an E∞ operad and let E be spectrum with an action of O. This action is
given by maps
ξj : O(j) nΣj E ∧j → E ,
making certain diagrams commute. Here n denotes twisted half-smash product in spectra. Let
W∗ be the standard free Cp -resolution of Fp with a single Fp [Cp ]-basis element ei in degree i. Let
C∗ (−) denote the cellular chains functor on CW-spaces or CW-spectra; there is then a chain map
W∗ → C∗ (O(p)) lifting the identity on Fp , unique up to homotopy. The image of ei under this
map we also denote by ei . Now let x ∈ Hq (E) be represented by a cycle z ∈ Cq (E), and consider
the image of ei ⊗ z ⊗p under the map
C∗ (O(p)) ⊗Σp C∗ (E)⊗p
∼
=
C∗ (O(p) nΣp E ∧p )
(ξp )∗
C∗ (E) ;
denote its image in homology by Qi (x). Then for p = 2 define Qi (x) = 0 when i < q, and
Qi (x) = Qi−q (x)
when i ≥ q. For p > 2 define Qi (x) = 0 when 2i < q, and
Qi (x) = (−1)i ν(q)Q(2i−q)(p−1) (x)
when 2i ≥ q, where ν(2`+) = (−1)` (m!) with m = 12 (p−1) and = 0, 1. See [BMMS86, Chapter
III] for more details.
An important source of E∞ ring spectra is the algebraic K-theory of symmetric bimonoidal
categories. The precise construction of the E∞ ring structure on the K-theory space or spectrum
of a bipermutative category has historically been a sticky point, with mistakes in several of the
older accounts. Here we will use the construction of Elmendorf-Mandell [EM06], but see also
[May09].
First let C be a bipermutative category. Here we use the term bipermutative category as originally defined in [May77], not the more general notion defined in [EM06]. From C, the construction
in [EM06] produces a symmetric spectrum in simplicial sets with an action of the simplicial BarrattEccles operad. After geometric realization this results in a symmetric spectrum in spaces which
we denote K(C). The operad action is given by maps
κj : EΣj nΣj K(C)∧j → K(C) ,
e j . Recall that the translation
where EΣj is the classifying space of the translation category Σ
category of a group (or even a set) G has as objects the elements in G, with a unique morphism
between each (ordered) pair of objects. The maps κj are induced by the bipermutative structure
in the following way. Let ⊗ denote the product in the bipermutative structure on C, and define a
functor
ej : Σ
e j nΣ C j → C
(2)
λ
j
on objects by sending (σ; a1 , . . . , aj ) to aσ−1 (1) ⊗ · · · ⊗ aσ−1 (j) . A morphism
(τ σ −1 ; f1 , . . . , fj ) : (σ; a1 , . . . , aj ) → (τ ; b1 , . . . , bj )
ej to the composition
is mapped by λ
aσ−1 (1) ⊗ · · · ⊗ aσ−1 (j)
Tσ
∼
=
f1 ⊗···⊗fj
a1 ⊗ · · · ⊗ aj
b1 ⊗ · · · ⊗ bj
Tτ
∼
=
bτ −1 (1) ⊗ · · · ⊗ bτ −1 (j) ,
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
5
where Tσ denotes the twist isomorphisms given by the bipermutative structure. Applying the
ej yields a map
classifying space functor B(−) to λ
λj : EΣj nΣj BC ∧j → BC .
The zeroth space in K(C) is BC, the classifying space of C; let Σ∞ BC → K(C) be the induced
map of symmetric spectra. The relationship between κj and λj is described by the commutative
diagram
κj
EΣj nΣj K(C)∧j
K(C)
(3)
EΣj nΣj (Σ∞ BC)∧j
∼
=
Σ∞ (EΣj nΣj BC ∧j )
Σ∞ λj
(Σ∞ BC) .
See [EM06, Section 8] for more details.
Now let C be a symmetric monoidal category. By [May77, VI.3.5] there is a natural equivalence
C → ΦC of symmetric monoidal categories, where ΦC is bipermutative. The K-theory of C is
defined to be the K-theory of ΦC, and by precomposing with the map BC → B(ΦC) we get that
(3) still holds for symmetric bimonoidal C.
Let G be a finite group, and let E G the category of finite G-sets and G-equivariant bijections.
This is a symmetric bimonoidal category under disjoint union and cartesian product. If we let
C = E G in (3), the maps λj are induced by the functors defined by
(4)
(σ; X1 , . . . , Xj ) 7→ Xσ−1 (1) × · · · × Xσ−1 (j)
on objects, and similarly for morphisms. The following result, proved in [BR10, 2.1], is essential
for the computation in Section 3.
Theorem 1.1. Let G be a finite group. There is an equivalence SG ' K(E G ) of spectra such that
the induced isomorphism H∗ (SG ) ∼
= H∗ (K(E G )) commutes with the Dyer-Lashof operations.
2. Topological cyclic homology
2.1. Topological Hochschild and cyclic homology of ring spectra. We recall Bökstedt’s
topological Hochschild homology construction. For published accounts with more details, see
[BHM93] and [HM97]. Let B be a symmetric ring spectrum, and let I denote Bökstedt’s category
of ... numbers and injective functions. Write i = (i0 , . . . , in ) ∈ I n+1 for elements in the product
category I n+1 . For each finite-dimensional T -representation V , define a space
(5)
thh(B; V )n = hocolim F (S i0 ∧ · · · ∧ S in , Bi0 ∧ · · · ∧ Bin ∧ S V ) ,
i∈I n+1
where the maps in the directed system The spaces thh(B; V )n assemble to a cyclic space thh(B; S V )· ,
i.e., a cyclic object in the category of spaces. Recall that a cyclic object is a contravariant functor
from the cyclic category Λ, which has the same objects as the simplicial category ∆, but an extra
“cyclic” operator tn : [n] → [n] satisfying certain relations including tnn = id. For thh(B; V )· , the
face and degeneracy maps are given by ..., and tn is given by cyclically permuting ...
The geometric realization thh(B; V ) = |thh(B; V )· | has a T -action coming from the cyclic
structure. It also has a T -action coming from the T -space S V , resulting in a T × T -action. The
restriction of this action to the diagonal T ⊂ T × T is the T -action we want on thh(B; V ). The
association V 7→ thh(B; S V ) is an equivariant T -prespectrum, with structure maps
S W ∧ thh(B; V ) → thh(B; W ⊕ V ) .
Let T HH(B) = L(thh(B)) be the associated T -spectrum.
In addition to being an equivariant T -spectrum, T HH(B) has the structure of a cyclotomic
spectrum. This is given by suitably compatible T -equivalences
rc : ρ∗C ΦC T HH(B) → T HH(B)
6
HÅKON SCHAD BERGSAKER
for each finite subgroup C ⊂ T . Here ρC : T → T /C is the |C|’th root isomorphism, and ρ∗C
pulls back a T /C-spectrum to a T -spectrum via ρC . The cyclotomic structure maps give rise to
restriction maps
R : T HH(B)D → T HH(B)C
for each pair of subgroups C ⊂ D ⊂ T with |C| dividing |D|. There are also the so-called Frobenius
maps
F : T HH(B)D → T HH(B)C
which by definition is the inclusion of fixed points.
Henceforward we will work at a prime p, and we only consider finite p-subgroups C ⊂ T . The
topological cyclic homology of B (at a prime p) can now be defined as
T C(B; p) = holim(. . .
R
T HH(B)Cpk
F
R
T HH(B)Cpk−1
F
R
...)
F
We will suppress the prime p in the notation and simply write T C(B) for T C(B; p).
2.2. Topological cyclic homology of spaces. Let X be a space, and B = S[ΩX]. In this
case we write T C(X) for T C(B). Note that for a topological monoid X, the spectrum S[ΩX] is
a commutative symmetric ring spectrum, so T HH(S[ΩX]) and T C(X) are E∞ ring spectra. A
homomorphism X → Y of monoids induces a map T C(X) → T C(Y ) of E∞ ring spectra.
An important result for us is the following, proved in [BHM93]. See also [Rog02, Section 1] for
clarifying points. Let ΛX denote the free loop space of X, and let ∆p : ΛX → ΛX be the map
that precomposes a loop S 1 → X with the degree p map S 1 → S 1 . The circle group T acts on ΛX
by rotating loops. There is a homotopy cartesian square
T C(X)
(6)
γ
Σ∞
+ ΛX
ΣΣ∞
+ (ΛX)hT
t
1−∆p
Σ∞
+ ΛX
,
where t is the dimension shifting transfer associated to the T -bundle ΛX → (ΛX)hT . The map is the natural map T C(X) → T HH(S[ΩX]) ' Σ∞
+ ΛX.
3. Topological cyclic homology of a point
Here we recall previous results about the multiplicative structure on the homology of T C(∗)
at the prime 2, and provide an argument for odd primes. We start off by describing the additive
structure of T C(∗).
If we let X = ∗ in (6), then Σ∞
+ ΛX ' S and the map ∆p is trivial. It follows that
(7)
∞
T C(∗) ' S ∨ hofib(t) ' S ∨ ΣCP−1
,
∞
∞
where hofib(t) is the fiber of the circle transfer t : ΣΣ∞
→ S, and CP−1
is the Thom spectrum
+ CP
∞
of the negative canonical complex line bundle over CP . The augmentation T HH(S) → S is an
equivalence of F-cyclotomic spectra (as is the unit map, see Lemma ??), and the map T C(∗) →
T HH(S) factors through a multiplicative map f : T C(∗) → SCp . By the Segal-tom Dieck splitting,
SCp ' S ∨ S[BCp ]; write
H∗ (SCp ) = Fp ⊕ Fp {αj | j ≥ 0}
for its homology, with |αj | = j. By (7) the homology of T C(∗) is
H∗ (T C(∗)) = Fp ⊕ Fp {Σβj | j ≥ −1} ,
with |Σβj | = 2j + 1. The induced map
(8)
f∗ : H∗ (T C(∗)) → H∗ (SCp )
in homology is given by f∗ (Σβj ) = α2j+1 in positive degrees. See [BR10, Section 1] for more
details.
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
7
The following is the main result of [BR10]. The action of the dual Steenrod operations is not
explicitly stated there, but can be extracted from the proof of [BR10, Theorem 0.2].
Theorem 3.1. Let p = 2. The dual Steenrod operations in H∗ (T C(∗)) are given by
2j − 2k + 1
2k
Sq∗ (Σβj ) =
Σβj−k .
2k
The Pontrjagin product is trivial, and the Dyer-Lashof operations are given by
i−1
2i
Q (Σβj ) =
Σβi+j , for i > j ≥ 0
j
Q0 (Σβ−1 ) = Σβ−1 .
The remaining operations are zero.
We will now prove the corresponding result for odd primes. The proof goes along the same lines
as for the prime 2, but extra care is needed to keep track of the behavior of the relevant elements
and maps in group homology. We follow [MM85] and utilize the Evens transfer map to describe
the homomorphisms that capture the multiplicative structure in homology. In fact Theorem 3.2
below should follow from [MM85, Theorem 6.8] and stabilization, which is a corresponding result
for the underlying E∞ ring space of SCp . However, they only state and prove the result for the
prime 2, noting that the argument for odd primes is similar. We find that the formulas for odd
primes become significantly more complicated; for this reason, and for completeness, we provide a
full proof.
Theorem 3.2. Let p be an odd prime. The dual Steenrod operations in H∗ (SCp ) are given by
[j/2] − k(p − 1)
k
P∗ (αj ) =
αj−2k(p−1)
k
β∗ (α2j ) = α2j−1 , β∗ (α2j+1 ) = 0 .
The Pontrjagin product is trivial, and the Dyer-Lashof operations are given by
i
i+j+1 i − 1
Q (αj ) = (−1)
αi(p−1)+j .
j
The remaining operations are zero.
Corollary 3.3. Let p be an odd prime. The dual Steenrod operations in H∗ (T C(∗)) are given by
j − k(p − 1)
k
Σβj−k(p−1) .
P∗ (Σβj ) =
k
The Pontrjagin product is trivial, and the Dyer-Lashof operations are given by
i
i+j+1 i − 1
Q (Σβj ) = (−1)
Σβi(p−1)+j , for i > j ≥ 0
j
Q0 (Σβ−1 ) = Σβ−1 .
The remaining operations are zero.
Proof. By Theorem 3.2 and the map (8), only the formulas involving Σβ−1 need verification. The
statements about dual Steenrod operations and the Pontrjagin product also easily follow directly,
∞
since T C(∗) ' S ∨ ΣCP−1
. Except for the unit summand, H∗ (T C(∗)) is concentrated in odd
degrees, so the product is automatically trivial. Formulas for the dual Steenrod operations follow
by the corresponding formulas for H∗ (CP ∞ ) and James periodicity. See [Rog03, Section 2] for
details.
For the Dyer-Lashof operations on Σβ−1 , the only option which is consistent with Theorem 3.2
and the Nishida relations is Q0 (Σβ−1 ) = Σβ−1 and Qi (Σβ−1 ) = 0 for all i ≥ 1. We refrain from
8
HÅKON SCHAD BERGSAKER
writing out the Nishida relations in their full generality, referring to [BMMS86, III.1] for more
details. The fact that Q0 (Σβ−1 ) = Σβ−1 follows from the relation
P∗p Qp−1 (Σβp−2 ) = Q0 P∗1 (Σβp−2 ) .
Here P∗p (Σβp2 −p−1 ) = Σβ−1 and P∗1 (Σβp−2 ) = Σβ−1 . Now the relation
N p
P∗i Qi (Σβ−1 ) = (−1)i N
Q0 P∗0 (Σβ−1 ) ,
p −i
where N is sufficiently large, implies that Qi (Σβ−1 ) = 0 for i ≥ 1.
As preparation for the proof of Theorem 3.2 we recall the Evens transfer map and some of its
properties, originally defined by Evens in [Eve63]. For a group H, write Σn o H for the wreath
product Σn n H ×n , with multiplication
(σ; h1 , . . . , hn ) · (τ ; h01 , . . . , h0n ) = (στ ; hτ (1) h01 , . . . , hτ (n) h0n ) .
Let H ⊂ G be a subgroup with finite index n = [G : H]. Choose coset representatives gi such that
(9)
n
[
G=
gi H .
i=1
For each g ∈ G and 1 ≤ i ≤ n, write ggi = gσg (i) hi (g), where σg ∈ Σn and hi (g) ∈ H. This defines
functions σ : G → Σn and hi : G → H. The Evens transfer
τ : G → Σn o H
is given by τ (g) = (σ(g); h1 (g), . . . , hn (g)); it is easily seen to be a group homomorphism. The
map τ depends on the choice of coset representatives, but only up to an inner automorphism of
Σn o H ×n .
Let AutG (X) denote the group of G-equivariant automorphisms of a (right) G-set X, and
consider the map f : AutG (G) → AutH (G) that forgets part of the equivariance. The coset
decomposition (9) specifies an isomorphism
(10)
Σn o H
∼
=
AutH (G)
given by sending (σ; h1 , . . . , hn ) to the H-automorphism of G which restricts to maps gi H → gσ(i) H
that acts by hi . Composing f with this isomorphism and the canonical identification G = AutG (G)
results in a map G → Σn o H, which is easily seen to coincide with the Evens transfer τ .
Now let G be a group with n elements and let ∆ ⊂ G×j be the diagonal subgroup, where j
is any positive integer. Fix an ordering G = {g1 , . . . , gn } of the elements of G. This induces the
lexicographic ordering on G×j . We let
(11)
τj : G×j → Σnj−1 o G
be the Evens transfer associated to the coset decomposition
[
G×j =
(1, gi2 , . . . , gij )∆ ,
1≤is ≤n
where we identify G with ∆.
For σ ∈ Σj , let aσ ∈ AutG (G×j ) be the automorphism that sends (x1 , . . . , xj ) to (xσ−1 (1) , . . . , xσ−1 (j) ).
It corresponds to an element cσ ∈ Σnj−1 o G under the isomorphism (10). We can now extend τj
to a map
ψj : Σj o G → Σnj−1 o G
by defining ψj (σ; x1 , . . . , xj ) = cσ τ (x1 , . . . , xj )c−1
σ ; we call it the extended Evens transfer. Let
φj : Σj o AutG (G)×j → AutG (G×j )
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
9
ej in (2). Thus
be the homomorphism defined similarly to morphism part of the functor λ
φj (σ; f1 , . . . , fj ) = aσ ◦ (f1 × · · · × fj ) ◦ a−1
σ .
Comparing the definitions we immediately get the following.
Lemma 3.4. Under the identification G = AutG (G) and the isomorphism (10), the map φj
coincides with the extended Evens transfer ψj .
From now on we will only consider the case G = Cp , and we also restrict ψj to Cj o Cp ⊂
Σj o Cp . Here the inclusion Cj ⊂ Σj is given by letting the generator c of Cj permute (1, . . . , j) as
(j, 1, . . . , j − 1). We define two homomorphisms needed for the next lemma. Let
µ : Cp×j → Σpj−1 × Cp
be defined by µ(x1 , . . . , xj ) = (σx , x1 ), where σx is the permutation sending (i2 , . . . , ij ) to (i2 +
x2 − x1 , . . . , ij + xj − x1 ). For the second, let τ ∈ Σpj−1 be the permutation given by sending
(i2 , . . . , ij ) to (i3 − i2 , . . . , ij − i2 , −i2 ). Now let
ν : Cj × Cp → Cp×p
j−1
be defined by ν(c; x) = (ν1 , . . . , νnj−1 ), where νi = x + i2 − τ −1 (i) for i = (i2 , . . . , ij ), and c ∈ Cj
is the generator.
Lemma 3.5. The diagram
Cp×j
(12)
i
Cj o Cp
µ
Σpj−1 × Cp
d
Cj × Cp
ν
ψj
d
Σpj−1 o Cp
i
Cp×p
j−1
commutes.
Proof. The first claim follows directly from the definitions. For the second, we only have to check
commutativity on elements of the form (c, x) ∈ Cj × Cp , where c ∈ Cj is the generator. To
identify the image ψj d(c, x) = bc τj (x, . . . , x)b−1
c , we first find the element bc . The automorphism
ac ∈ AutCp (Cp×j ) maps the coset (1, i2 , . . . , ij ) + ∆ to (1, i3 − i2 , . . . , ij − i2 , −i2 ) + ∆ by adding i2
to each coordinate. Hence under the isomorphism (10) it corresponds to the element
bc = (τ ; 0, . . . , 0, 1, . . . , 1, . . . , p − 1, . . . , p − 1) ∈ Σpj−1 o Cp ,
each number repeated pj−2 times. Here τ is the permutation used in the definition of ν above.
Using the definition of τj , we have τj (x, . . . , x) = (1; x, . . . , x).
The following lemma is the main calculation we need for the proof of Theorem 3.2. Before we
state it, we recall some standard group homological facts. Recall that H∗ (Cp ) = Fp {αj | j ≥ 0}
and
H∗ (Σp ) = Fp {αj | j = 2k(p − 1) or j = 2k(p − 1) − 1, k ≥ 0} ,
with the induced map H∗ (Cp ) → H∗ (Σp ) being the canonical projection. For any π ⊂ Σj , with j
a positive integer, we have
H∗ (π o Cp ) ∼
= H∗ (π; H∗ (Cp )⊗j ) .
When π = Cp , the Cp -module H∗ (Cp )⊗p can be decomposed as
H∗ (Cp )⊗p ∼
= A ⊕ (Fp [Cp ] ⊗ B) ,
where A = Fp {αj⊗p | j ≥ 0} and B = Fp {αγ(j1 ) ⊗ · · · ⊗ αγ(jp ) | γ ∈ C, js ≤ js+1 , j1 < jp }. Here C
is a set of coset representatives for Cp in Σp . This implies that
H∗ (Cp ; H∗ (Cp )⊗p ) ∼
= (H∗ (Cp ) ⊗ A) ⊕ B .
10
HÅKON SCHAD BERGSAKER
For the map
j∗ : H∗ (Cp ; H∗ (Cp )⊗p ) → H∗ (Σp ; H∗ (Cp )⊗p )
(13)
induced by the inclusion Cp ⊂ Σp , j∗ (αi ⊗ αj⊗p ) is zero unless
(i) j is even and i = 2t(p − 1) or i = 2t(p − 1) − 1.
(ii) j is odd and i = (2t + 1)(p − 1) or (2t + 1)(p − 1) − 1.
See [May70] for more details.
The classifying space BE Cp has the structure of an (additive) E∞ space coming from disjoint
union of Cp -sets, and as such has Dyer-Lashof operations in homology which we denote by Q`a , to
differentiate them from the multiplicative operations in H∗ (K(E Cp )).
Lemma 3.6.
(i) The induced map
(ψ2 )∗ : H∗ (C2 o Cp ) → H∗ (BE Cp )
satisifies
(ψ2 )∗ (1 ⊗ αj ⊗ αk ) =
X
c` Q`a (αj+k−2`(p−1) ) + d` βQ`a (αj+k−2`(p−1)+1 ) ,
`
for some constants c` and d` .
(ii) The induced map
(ψp )∗ : H∗ (Cp o Cp ) → H∗ (Σpp−1 o Cp )
satisfies
(ψp )∗ (ei ⊗ αs⊗p ) =
X
Proof. For (i), we use the left-hand square in (12) with j = 2. In this case µ is given by µ(c1 , c2 ) =
(c2 − c1 , c1 ), where c2 − c1 ∈ Cp ⊂ Σp is considered as a cyclic permutation. In homology we now
have the commutative diagram
H∗ (Cp × Cp )
i∗
H∗ (C2 o Cp )
µ∗
H∗ (Σp × Cp )
(ψ2 )∗
d∗
H∗ (Σp o Cp ) ,
and to prove the stated formula we evaluate the effect of d∗ µ∗ on the element αj ⊗ αk . Since µ
takes values in Cp × Cp , the composition d ◦ µ factors as
Cp × Cp
µ
Cp × Cp
d
Cp o Cp
j
Σp o Cp .
In homology we have
µ∗ (αj ⊗ 1) = (χ ⊗ 1) ◦ ∆(αj ) =
j
X
(−1)j−s αj−s ⊗ αs ,
s=0
where χ and ∆ are the conjugation and coproduct in H∗ (Cp ), respectively. Combined with the
formula µ∗ (1 ⊗ αk ) = αk ⊗ 1, this yields
j
j
X
X
j+k−s
µ∗ (αj ⊗ αk ) =
(−1)j+(k−1)s (αj−s ∗ αk ) ⊗ αs =
(−1)j+(k−1)s
αj+k−s ⊗ αs ,
k
s=0
s=0
where ∗ denotes the Pontrjagin product in H∗ (Cp ). A standard computation gives a formula for
d∗ in terms of the dual Steenrod operations, see e.g. [May70, 9.1]. With our notation it is given
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
11
by
(14)
d∗ (αj ⊗ αk ) = ν(k)
X
(−1)t αj+(2pt−k)(p−1) ⊗ P∗t (αk )⊗p
t
− δ(j)ν(k − 1)
X
(−1)t αj+p+(2pt−k)(p−1) ⊗ P∗t β∗ (αk )⊗p ,
t
`
where ν(2` + ) = (−1) (m!) , m = (p − 1)/2, and δ(2` + ) = . Using the explicit formulas for
the P∗t , we have
j X
X
⊗p
(15) d∗ µ∗ (αj ⊗ αk ) =
c(s, t)ν(s)αj+k−s+(2pt−s)(p−1) ⊗ αs−2t(p−1)
t
s=0
⊗p
− δ(s − 1)δ(j + k − s)d(s, t)ν(s − 1)αj+k−s+p+(2pt−s)(p−1) ⊗ αs−1−2t(p−1)
,
where
c(s, t) = (−1)
j+(k−1)s+t
j + k − s [s/2] − t(p − 1)
k
t
and
j + k − s [(s − 1)/2] − t(p − 1)
.
k
t
Since we are really interested in the image in H∗ (Σp o Cp ), i.e., after composing with j∗ , the
degrees of the α-elements are subject to the constraints in (13). We consider two cases, according
to the parity of j + k. If j + k is even, all the terms on the second line in (15) vanish because of the
δ-factors. For the remaining terms, we first consider the terms with s even. Then αs−2t(p−1) is in
even degree and 2(p − 1) divides j + k − s + (2pt − s)(p − 1), hence we can write j + k − s = 2i(p − 1)
for some i. For odd s,
j + k − s + (2pt − s)(p − 1) = u(p − 1) − 1
d(s, t) = (−1)j+(k−1)s+t
with u odd, hence j + k − s = 2i(p − 1) − 1 for some i. Letting ` = i + t, and noting that
ν(s + 4v) = ν(s), we can rewrite the sum (15) for j + k even, after application of j∗ , as
X
⊗p
j∗ d∗ µ∗ (αj ⊗ αk ) =
c` (−1)` ν(j + k − 2`(p − 1))α(2`−(j+k−2`(p−1))(p−1) ⊗ αj+k−2`(p−1)
`
⊗p
+ d` (−1)` ν(j + k − 2`(p − 1))α(2`−(j+k−2`(p−1))(p−1) ⊗ αj+k−2`(p−1)+1
where
c` =
X
j+t
(−1)
t
2t(p − 1)
j + k − 2t(p − 1)
(j + k)/2 − `(p − 1)
`−t
and
d` =
X
t
(−1)
j+t
2t(p − 1)
j + k − 2t(p − 1)
(j + k)/2 − `(p − 1)
.
`−t
For (ii), choose j = p in the right-hand square in (12).
Proof of Theorem 3.2. The idea of the proof is to use the equivalence in Theorem 1.1 and study
the maps of Cp -sets inducing the multiplicative structure on K-theory. This in turn entails finding
explicit formulas in homology for certain group homomorphisms. The problem is to describe these
group homomorphisms in a convenient way so as to be able to extract the relevant information.
Fortunately we have already done all the hard work.
Consider the map
λj : EΣj nΣj (BE Cp )∧j → BE Cp
12
HÅKON SCHAD BERGSAKER
defined in (4). The classifying space BE Cp decomposes as a disjoint union of components
a
B Aut(X) ,
BE Cp '
[X]
where the union runs over isomorphism classes of finite Cp -sets. Let ιj : B Aut(Cp×j ) → BE Cp
denote the inclusion of the component of X = Cp×j . We have the commutative diagram
λj
EΣj nΣj (BE Cp )∧j
BE Cp
ιj
1nι∧j
1
EΣj nΣj (B Aut(Cp ))∧j
∼
=
B(Σj o Aut(Cp ))
φ
Σj o Aut(G)
Bφ
B Aut(Cp×j ) ,
Aut(G×j )
∼
=
Cj o G
ψ
Σnj−1 o G
4. Topological cyclic homology of the circle
Here we compute the homology and induced multiplicative structure of T C(S 1 ). As in the case
T C(∗), the result will follow from a computation of the corresponding structure for the second
stage in the inverse system defining T C. Since T C(S 1 ) = T C(S[Z]), this inverse system consists
of cyclic fixed points of T HH(S[Z]). To ease the notation we will often write T (−) for T HH(−).
4.1. The homology of T (S[Z])Cp . First we need some general results about T HH of group rings.
Recall that a topological monoid G is called group-like if π0 (G) is a group.
Proposition 4.1. Let G be a topological group-like monoid.
(i) Let A be a symmetric ring spectrum. There is an equivalence of cyclotomic spectra
T (A[G]) ' T (A) ∧ S[B cy (G)] .
(ii) There is an equivalence of cyclotomic spectra
S[ΛBG] ' S[B cy (G)] .
If G is commutative, both equivalences are equivalences of E∞ ring spectra.
The inclusion of constant loops BG → ΛBG induces a map T (A) ∧ BG+ → T (A) ∧ ΛBG+ ,
which combined with the equivalence in Proposition 4.1 gives the assembly map
aT : T (A) ∧ BG+ → T (A[G]) .
The inclusion BG → ΛBG is T -equivariant when we give BG the trivial action, and the map aT
induces an assembly map
(16)
aT C : T C(A) ∧ BG+ → T C(A[G]) .
We now specialize to the case A = S, G = Z.
Lemma 4.2. The unit map
ι : S → T (S)
is a map of cyclotomic spectra which is a C-equivalence for all finite subgroups C ⊂ T .
Combining 4.1 and 4.2 we get the following.
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
13
Corollary 4.3. There is an E∞ ring map of cyclotomic spectra
S[ΛS 1 ] → T (S[Z])
which is a C-equivalence for all finite subgroups C ⊂ T .
Additively, the fixed point spectra T (S[Z])Cpk ' S[ΛS 1 ]Cpk can be identified by applying the
Segal-tom Dieck splitting. We will need a stronger multiplicative statement, which is a reformulation of a result by Hesselholt and Madsen [HM04].
Proposition 4.4. There is an E∞ ring map f which is part of a split cofiber sequence
Wk W
f
Cpk−s
∗
T (A)Cpk [ΛS 1 ]
T (A)[ΛS 1 ]Cpk
∧ S 1 (n)+ )Cps .
s=1 p-n (ρ T (A)
We turn to the homological analysis. First note that H∗ (S[Z]) = Fp [x, x−1 ], concentrated in
degree zero. If we forget the T -action on ΛS 1 , we have an equivalence of topological groups
ΛS 1 ' Z × S 1 .
Hence we get an equivalence of E∞ ring spectra S[ΛS 1 ] ' S[Z] ∧ S[S 1 ], with homology algebra
H∗ (S[ΛS 1 ]) = Fp [x, x−1 ] ⊗ E(e) .
Here e is in degree 1. The action of the dual Steenrod algebra on H∗ (S[ΛS 1 ]) is trivial, and the
only non-trivial Dyer-Lashof operation is Q0 (xk ) = xpk .
Next we consider the homology of the fixed point spectrum S[ΛS 1 ]Cp . Proposition 4.4 immediately implies the following result.
Proposition 4.5. Let p be any prime.
(1) There is a split short exact sequence
H∗ (SCp ) ⊗ P (x±1 ) ⊗ E(e)
0
f∗
H∗ (S[ΛS 1 ]Cp )
Fp {e0 (n), e1 (n) | n ∈ Z}
0,
of right A∗ -modules, with |x| = 0, |e| = 1 and |es (n)| = s. The map f∗ is an inclusion of
algebras.
(2) The dual Steenrod operations in H∗ (S[ΛS 1 ]Cp ) are trivial on x±1 , e and es (n). The action
on the H∗ (SCp ) factor is given by Theorem 3.2.
(3) The Dyer-Lashof operations in H∗ (S[ΛS 1 ]Cp ) are trivial on x±1 and e, with the exception
that Q0 (xk ) = xpk . The action on the H∗ (SCp ) factor is given by Theorem 3.2.
4.2. The homology of T C(S 1 ). Here we use the defining cartesian square (6) to calculate the
homotopy type and homology of T C(S 1 ). As a T -space
a
(17)
ΛS 1 '
S 1 (n) ,
n∈Z
where S 1 (n) is S 1 with a T -action given by t · s = tn s. Write
M
H∗ (ΛS 1 ) =
Fp {e0 (n), e1 (n)}
n∈Z
for its homology, where ei (n) is in degree i. The map
σ∗ : H∗ (ΛS 1 ) → H∗+1 (ΛS 1 )
(18)
induced from the T -action is given by σ∗ (e0 (n)) = ne1 (n). We now have
a
a
a
(ΛS 1 )hT '
ET ×T S 1 (n) '
BCn
CP ∞ × S 1 ,
n∈Z
n6=0
where we define Cn = C−n for negative n, and we can easily write down the homology
M
M
M
(19)
H∗ ((ΛS 1 )hT ) =
Fp {α0 (n)}
Fp {αj (n) | j ≥ 0}
Fp {βj , βj0 | j ≥ 0} .
p-n
p|n6=0
14
HÅKON SCHAD BERGSAKER
Here αj (n) has degree j, βj has degree 2j and βj0 has degree 2j + 1. We have chosen the generators
so that
(
α0 (n) if n 6= 0
,
(20)
π∗ (e0 (n)) =
β0
if n = 0
where π : ΛS 1 → (ΛS 1 )hT is the canonical projection. Note that the summands with p - n reduce
to just an Fp in degree zero, or in other words, (BCn )∧
p ' ∗.
Lemma 4.6. The induced map in homology
t∗ : H∗ (Σ(ΛS 1 )hT ) → H∗ (ΛS 1 )
of the circle transfer is given by
t∗ (Σα0 (n)) = ne1 (n)
t∗ (Σβ0 ) = 0 .
The remaining classes are mapped to zero by degree reasons.
Proof. We will use the “degenerate double coset formula” from [MMM86, Lemma 2.7], which for
us takes the form of a commutative diagram
Σπ
1
S 1 ∧ Σ∞
+ ΛS
1
ΣΣ∞
+ (ΛS )hT
[T ]∧1
t
α
1
T+ ∧ Σ∞
+ ΛS
1
Σ∞
+ ΛS .
Here π is the projection, [T ] ∈ π1s (T+ ) is the fundamental class, and α is the action map. Applying
homology we get the following diagram.
Σπ∗
ΣH∗ (ΛS 1 )
ΣH∗ ((ΛS 1 )hT )
t∗
[T ]∧1
α∗
H∗ (T ) ⊗ H∗ (ΛS 1 )
H∗ (ΛS 1 ) .
Here α∗ restricted to ΣH∗ (ΛS 1 ) ⊂ H∗ (T ) ⊗ H∗ (ΛS 1 ) coincides with the map σ∗ in (18). Tracing
the elements Σe0 (n) ∈ ΣH∗ (ΛS 1 ) around the diagram, using (20) and (18), gives the result.
Proposition 4.7. There is an equivalence of spectra
1
T C(S ) ' T C(∗) ∧
1
S+
∨
_
p-n
S
−1
∨
∞
_
!
ΣΣ∞
+ BCpk
k=0
1
such that the inclusion of the summand T C(∗) ∧ S+
coincides with the assembly map.
Proof. Using the splitting (17) of ΛS 1 , the homotopy cartesian square (6) with X = S 1 splits as
a wedge of simpler squares. Corresponding to n = 0 is the square
∞
1
ΣΣ∞
∧ S+
+ CP
1
T C(∗) ∧ S+
t
1
Σ∞
+ S (0)
1−∆p
1
Σ∞
+ S (0) ,
HOMOLOGY OPERATIONS IN THE TOPOLOGICAL CYCLIC HOMOLOGY OF THE CIRCLE
15
1
which is the square for T C(∗) smashed with S+
. Note that 1 − ∆p ' ∗ in this case. For each n 6= 0
with p - n, there is a pullback square
W∞
∞
C
k=0 ΣΣ+ BCpk n
(21)
t
W∞
k=0
1 k
Σ∞
+ S (p n)
1−∆p
W∞
k=0
1 k
Σ∞
+ S (p n) .
To identify C we use the split cofiber sequence
W∞
k=0
1−∆p
1 k
Σ∞
+ S (p n)
W∞
k=0
1 k
Σ∞
+ S (p n)
∇
1
Σ∞
+S ,
where ∇ is the identity on each summand. This gives a splitting of the bottom right corner in
(21), and induces a splitting of C as the sum of the pullbacks in the two squares
S −1
ΣΣ∞
+ BCn
t
W∞
k=0
1 k
Σ∞
+ S (p n)
∗
1
Σ∞
+S
and
W∞
k=0
ΣΣ∞
+ BCpk
W∞
1 k
Σ∞
+ S (p n)
W∞
k=0
ΣΣ∞
+ BCpk n
t
W∞
k=0
1
k=0
1 k
Σ∞
+ S (p n) .
Here we use that BCpk n ' BCpk after p-completion. We can now conclude that C ' S −1 ∨
W
∞
∞
k=0 ΣΣ+ BCpk .
In conclusion we have the following description of the homology of T C(S 1 ).
Theorem 4.8. Let p be any prime.
(1) The homology of T C(S 1 ) is
H∗ (T C(S 1 )) = Fp {a−1 (m), 1, a0 , e1 , as (n) | m = 0 or p - m, s ≥ 1, n ∈ Z} ,
with subscripts indicating degree. The Pontrjagin product is trivial, with 1 being the unit.
(2) The dual Steenrod operations in H∗ (T C(S 1 )) are given by
s−k
Sq∗k (as (n)) =
as−k (n)
k
for p = 2, and
[s/2] − k(p − 1)
aj−2k(p−1) (n)
k
β∗ (a2j (n)) = a2j−1 (n) if p - n
P∗k (as (n)) =
for odd p. The remaining operations are trivial.
(3) The Dyer-Lashof operations in H∗ (T C(S 1 )) are given by
i
i+s+1 i − 1
Q (as (n)) = (−1)
as+i(p−1) (pn)
s
Q0 (a−1 (0)) = a−1 (0) .
The remaining operations are trivial.
16
HÅKON SCHAD BERGSAKER
Proof. All the statements involving only elements in non-negative degree will follow from Proposition 4.5, once we have determined the map H∗ (T C(S 1 )) → H∗ (S[ΛS 1 ]Cp ) in homology.
For the remaining formulas involving a−1 (m), we first use the map
H∗ (T C(∗)) → H∗ (T C(S 1 ))
induced from the inclusion of a point in S 1 . This map respects all the structure in question and
maps Σβ−1 to a−1 (0), so the formulas for a−1 (0) follow from Theorem 3.1 and Corollary 3.3.
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E-mail address: hakonsb@math.uio.no