TATE COHOMOLOGY OF CIRCLE ACTIONS AS A
HEISENBERG GROUP
JACK MORAVA
Abstract.We study the Madsen-Tillmann spectrum CP1-1as a quotient of
the Mahowald pro-object CP1-1, which is closely related to the Tate coho-
mology of circle actions. That theory has an associated symplectic struc*
*ture,
whose symmetries define the Virasoro operations on the cohomology of mod*
*uli
space constructed by Kontsevich and Witten.
1.Tate cohomology of circle actions
1.1 If E is a geometric bordism theory (such as integral homology), its Tate co-
homology t*TE can be constructed by tom Dieck stabilization from the geometric
theory of E-manifolds with T-action, the action required to be free on the boun*
*dary
[8]. If E is multiplicative, so is t*TE; there is a cofibration sequence
. .!.E*BT+ ! t*TE ! E-*-2BT+ ! . . .
in which the boundary map sends a T-manifold with boundary to the quotient of i*
*ts
boundary by the (free) T-action. When E is complex-oriented [eg MU or HZ] this
sequence reduces to a short exact sequence which identifies t*TE with the Laure*
*nt
series ring E*((e)) obtained by inverting the Euler class in E*(BT) = E*[[e]],
and the boundary map can be calculated as a formal residue; more precisely, the
formal Laurent series f maps to the residue of fd logEat e = 0, where d logEis
the invariant differential of the formal group law of E. When E is not complex-
orientable, tTE can behave very differently [4], as the Segal conjecture shows:*
* up
to a profinite completion,
Y
tTS0 ~ S0 _ S1 BT=C
where C runs through proper subgroups of T.
There is a related but simpler theory ø*TE defined by manifolds with free T-act*
*ion
on the boundary alone, which fits in an exact sequence
. .!.E* ! ø*TE ! E-*-2BT+ ! . .;.
ignoring the interior T-action defines a truncation
t*TE ! ø*TE .
____________
Date: 15 September 2001.
1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx.
The author was supported in part by the NSF.
1
2 JACK MORAVA
1.2 It is natural to think of t*TE as the E-homology of a version [1] of Mahowa*
*ld's
pro-spectrum CP-11, constructed from the inverse system
{CP-1k= T h(-kj)}
of Thom spectra defined by the filtered vector bundle
. . .kj (k + 1)j . . .
over CP 1. To be more precise,
tTE ~ E ^ S2CP-11
as spectra, and there is a similar equivalence
øTE ~ E ^ S2CP-11.
From this point of view, the morphism from t*TE to ø*TE is the E-homology of the
collapse map
CP-11 ! CP-11.
The cobordism class defined by a family of complex-oriented surfaces with a free
circle action on its boundary, parametrized by X, defines an element of
ø-2TMU(X+ ) = [X+ , CP-11^ MU] ;
the image of this class under the Thom homomorphism from MU to HZ is the
homomorphism
H*(CP-11, Z) ! H*(X+ , Z)
defined by the classifying map of Madsen and Tillmann, sketched below.
1.3 When E = HZ, the symmetric bilinear form
f, g 7! (f, g) = rese=0(fg de)
on the Laurent series ring tTHZ is nondegenerate, and the involution I(z) = z-1
on T defines a symplectic form
{f, g} = (I(f), g)
which restricts to zero on the subspaces of elements of degree 0 and < 0. The
Tate cohomology thus has an intrinsic inner product, with canonical polarization
and involution.
The functor ~Gwhich sends the commutative ring A to the set of formal Laurent
series
X p
G~(A) = {g = gkxk+1 2 A((x)) | g0 2 Ax , gk 2 A if0 > k}
k 0
(i.e. with g0 a unit and gk nilpotent when k is negative) in fact takes values*
* in
the category of groups, with formal composition of series as the operation. Th*
*is
group of invertible `nil-Laurent' series has a linear representation on the abe*
*lian
group valued functor A 7! A((x)), but it is a little too large to be convenient*
*ly
representable; in a certain sense it is an ind-pro-algebraic analog of the grou*
*p of
diffeomorphisms of the circle.
TATE COHOMOLOGY OF CIRCLE ACTIONS AS A HEISENBERG GROUP 3
1.4 It is tempting to interpret ~Gas a group of automorphisms of the Tate coho-
mology, but the most obvious action does not preserve the symplectic structure.
Kontsevich-Witten theory suggests a better alternative: there is an embedding
p
ek 7! fl-k-1_(x) : t*THZ ! R(( x))
2
of symplectic modules, defined using the fractional divided power
s
fls(x) = ___x___ (1,+ s)
in which the symplectic structure on the target is defined by
u, v 7! {u, v} = resx=0udv .
[The reals are a notational convenience: some powers of ß have been ignored.]
Over any field of characteristic zero, the square root ofpan invertible nil-Lau*
*rent
series in x is an invertible odd nil-Laurent series in x, and it makes better*
* sense
to think ofpthe group ~G-1=2of such series [8 x1.3] as symplectic automorphisms*
* of
t*THQ R(( x)). The half-integral shift comes ultimately from the fact that t*
*THZ
is not Spanier-Whitehead self-dual; rather, its dual is most naturally interpre*
*ted as
its own double suspension.
2.Madsen-Tillmann and Kontsevich-Witten
Madsen and Tillmann construct a map
a
BDiff(Fg) ! 1 CP-11
g 0
which is compatible with gluing of surfaces; in particular, it defines a lax fu*
*nctor
from the two-dimensional topological gravity category [10] to a topological cat*
*egory
with one object and an H-space of morphisms. [Reversing the orientation of a
surface corresponds to the involution I.] The point of this note is to identif*
*y a
suitable subgroup of ~G-1=2as the motivic automorphisms of this functor.
2.1 Here is a quick account of one component of [7]: if F Rn is a closed two-
manifold embedded smoothly in a high-dimensional Euclidean space, its Pontrjagi*
*n-
Thom construction Rn+! F maps compactified Euclidean space to the Thom
space of the normal bundle of the embedding. The tangent plane to F is classifi*
*ed
by a map ø : F ! Grass2,nto the Grassmannian of oriented two-planes in Rn, and
the canonical two-plane bundle j over this space has a complementary (n-2)-plane
bundle, which I will call (n - j). The normal bundle is the pullback along ø *
*of
(n - j); composing the map induced on Thom spaces with the collapse defines
Rn+! F ! Grass(n-j)2,n.
The space Emb (F ) of embeddings of F in Rn becomes highly connected as n in-
creases, and the group Diff(F ) of orientation-preserving diffeomorphisms of F *
*acts
freely on it, defining a compatible family
Rn+^DiffEmb(F ) ! Grass(n-j)2,n
which can be interpreted as a morphism
BDiff(F ) ! lim nGrass(n-j)2,n:= 1 CP-11.
4 JACK MORAVA
2.2 Madsen and Tillmann show their construction factors through an infinite loo*
*pspace
map
Z x B +1! 1 CP-11! Q(CP+1) ,
in which the last arrow is defined by collapsing the bottom two-cell in a cofib*
*ration
S-2 ! CP-11! CP+1 .
The fiber 2QS0 of the induced map of loop spaces is torsion, so the rational
cohomology of 1 CP-11is isomorphic to the algebra of symmetric functions on
the subspace of non-negative powers in t*THQ. This algebra is thus canonically
isomorphic to the Fock representation [10 x2.2] of the Heisenberg algebra of th*
*at
symplectic module; but this representation possesses a canonical Virasoro actio*
*n,
defining a homomorphism
H*(Z x B +1, Q) ! Symm (H*(CP+1)) 2 (G~-1=2- representations) .
In Kontsevich-Witten theory the usual generators bk 2 H*(CP+1), k 0, map to
symmetric functions
Tracefl-k-1_( 2) ~ -(2k - 1)!! Trace -2k-1 = tk( )
2
of a positive-definite Hermitian matrix ; this leads to a construction of the *
*appro-
priate twisted Virasoro representation in terms of Schur Q-functions [2,6].
2.3 The homomorphism
limMU*+n-2(T h(n - j)) ! MU*-2(BDiff(F ))
defined on cobordism by the Madsen-Tillmann construction sends the Thom class
to a kind of Euler class: according to Quillen's conventions, the Thom class is*
* the
zero-section of (n - j), regarded as a cobordism class of maps between manifold*
*s.
Its image is the class defined by the fiber product
Zn _____________//Grass2,n
| |
| |
fflffl| fflffl|
Rn+^DiffEmb(F )_____//Grass(n-j)2,n;
it is the space of equivalence classes, under the action of Diff(F ), of pairs *
*(x, OE),
with x 2 OE(F ) Rn a point of the surface (ie, in the zero-section of ), and*
* OE an
embedding. The image is thus the element
[Zn ! Rn+^DiffEmb] 7! MUn-2(SnBDiff(F ))
defined by the tautological family F xDiffEDiff(F ) of surfaces over the classi*
*fying
space of the diffeomorphism group. This class is primitive in the Hopf-like str*
*ucture
defined by gluing [9 x2.2], so the class
= exp(th(-j)v) 2 MU0Q( 1 CP-11)[[v]]
of finite unordered configurations of points on the universal surface (with v a*
* book-
keeping indeterminate) defines a multiplicative transformation
~ *: H*(Q(CP+1), Q) ! H*(MU, Q[[v]])
TATE COHOMOLOGY OF CIRCLE ACTIONS AS A HEISENBERG GROUP 5
with properties analogous to the Chern character of a vector bundle. It sends t*
*he
Fock representation described above to an algebra of cohomological characterist*
*ic
numbers.
2.4 Kontsevich-Witten theory uses`a more_sophisticated configuration space, whi*
*ch
maps the rational homology of Q( g 0M g) (suitably interpreted, for small g) t*
*o a
similar ring of characteristic numbers; this homology contains a fundamental cl*
*ass
a __ X __
[Q( M g)] = exp( [M g]v3(g-1))
g 0
for the moduli space of not necessarily connected curves. Witten's tau-function*
* is
the image of this `highest-weight' vector under the analog of ~ *; it is invari*
*ant under
the subalgebra of Virasoro generated by operators Lk of cohomological degree k,
with k -1.
3.Afterthoughts, and possible generalizations
3.1 Witten has proposed a generalization of 2D topological gravity which encom-
passes surfaces with higher spin structures: for a closed smooth surface F an r*
*-spin
structure is roughly a complex line bundle L together with a fixed isomorphism
L r ~=TF of two-plane bundles, but for surfaces with nodes or marked points the
necessary technicalities are formidable [5]. The group of automorphisms of such*
* a
structure is an extension of its group of diffeomorphisms by the group of rth r*
*oots
of unity, and there is a natural analog of the group completion of the category
defined by such surfaces. The generalized Madsen-Tillmann construction maps this
loopspace to the Thom spectrum T h(-jr), and it is reasonable to expect that th*
*is
map is equivariant with respect to automorphisms of the group of roots of unity.
This fits with some classical homotopy theory: if (for simplicity) r = p is pri*
*me,
multiplication by an integer u relatively prime to p in the H-space structure of
CP 1 defines a morphism
T h(-jp) ! T h(-jup)
of spectra, and the classification of fiber-homotopy equivalences of vector bun*
*dles
yields an equivalence of T h(-jup) with T h(-jp) after p-completion. There is
an analogous decomposition of tTHZp and a corresponding decomposition of the
associated Fock representations [10 x2.4].
3.2 In an extension of her work Tillmann also considers categories of surfaces
mapped to some parameter space X, which has interesting connections with both
Tate and quantum cohomology. When X is a smooth compact almost-complex
manifold, its Hodge-deRham cohomology admits a natural action of the Lie algebra
sl2(R), generated [12 IV x4] by the Hodge dimension operator H, multiplication *
*by
the first Chern class c1(X) = E, and its adjoint F = *E*. Recently Givental [3
x8.1] has shown that earlier work of (the schools of) Eguchi, Dubrovin, and oth*
*ers
on the Virasoro structure of quantum cohomology can be formulated in terms of
(what I like to think of as) t*TH(X, R), polarized by the twisted involution
1_H -1_H
IGiv= e2 e-E IeE e 2 .
It would be very interesting if this polarization could be understood in terms *
*of the
equivariant geometry of the free loopspace of X.
6 JACK MORAVA
3.3 I owe thanks to R. Cohen, E. Getzler, A. Givental, J. Greenlees, I. Madsen,
N. Strickland, and U. Tillmann (at least), for their forebearance in the face o*
*f my
continued misunderstanding of things they have tried patiently to tell me. I ho*
*pe
I'm finally starting to get it right.
References
1. R.L. Cohen, J.D.S. Jones, G.B. Segal, Floer's infinite dimensional Morse the*
*ory and homotopy
theory, in the Floer Memorial Volume, Birkhäuser, Progress in Mathematics 13*
*3 (1995)
297-326
2. P. DiFrancesco, C. Itzykson, J.-B. Zuber, Polynomial averages in the Kontsev*
*ich model, CMP
151 (1993) 193-219
3. A. Givental, Gromov - Witten invariants and quantization of quadratic hamilt*
*onians, available
at math.AG/0108100
4. J. Greenlees, A rational splitting theorem for the universal space for almos*
*t free actions, Bull.
London Math. Soc. 28 (1996) 183-189
5. T. Jarvis, T. Kimura, A. Vaintrob, Moduli spaces of higher spin curves and i*
*ntegrable hierar-
chies, available at math.AG/9905034
6. T. J'ozefiak, Symmetric functions in the Kontsevich-Witten intersection theo*
*ry of the moduli
space of curves, Lett. Math. Phys. 33 (1995) 347-351
7. I. Madsen, U. Tillmann, The stable mapping-class group and Q(CP1+), Aarhus p*
*reprint 14
(1999); and further work in in progress
8. J. Morava, Cobordism of involutions revisited, revisited, in the Boardman Fe*
*stschrift, Con-
temporary Math. 239 (1999)
9. __-, Topological gravity in dimensions two and four, available at math.QA/99*
*08006
10.__-, An algebraic analog of the Virasoro group, Czech. J. Phys. 51 (2001)
11.__-, A rudimentary theory of topological 4D gravity, Adv. Th. and Math. Phys*
*. (to appear);
available at math.DG/0007018
12.J.P. Serre, Algebres de Lie semisimples complexes, Benjamin (1966)
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: jack@math.jhu.edu