HEISENBERG GROUPS AND ALGEBRAIC TOPOLOGY 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.Introduction A sphere Sn maps essentially to a sphere Sk only if n k, and since we usually think of spaces as constructed by attaching cells, it follows that algebraic to* *pology is in some natural sense upper-triangular, and thus not very self-dual: as in t* *he category of modules over the mod p group ring of a p-group, its objects are bui* *lt by iterated extensions from a small list of simple ones. Representation theorists find semi-simple categories more congenial, and for re* *lated reasons, physicists are happiest in Hilbert space. This paper is concerned with* * some remarkable properties of the cohomology of the moduli space of Riemann surfaces discovered by physicists studying two-dimensional topological gravity (an enorm* *ous elaboration of conformal field theory), which appear at first sight quite unfam* *iliar. Our argument is that these new phenomena are forced by the physicists' interest in self-dual constructions, which leads to objects which are (from the point of* * view of classical algebraic topology) very large [1 x2]. Fortunately, equivariant homotopy theory provides us with tools to manage these constructions. The first section below is a geometric introduction to the Tate cohomology of the circle group; the conclusion is that it possesses an intrinsic symplectic module structure, which pairs positive and negative dimensions in a way very useful for applications. Section two studies operations on this (not q* *uite cohomology) functor, and exhibits the action of an algebraic analog of the Vira* *soro group on it. The third section relates rational Tate cohomology of the circle * *to that of the infinite loopspace QCP+1 considered by Madsen and Tillmann in recent work on Mumford's conjecture. I owe thanks to many people for help with the ideas in this paper, but it is es* *sentially a collage of a lifetime's conversations with Graeme Segal, who more or less ado* *pted me when we were both very young. ____________ Date: 6 January 2003. 1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx. The author was supported in part by the NSF. 1 2 JACK MORAVA 2. Geometric Tate cohomology 2.1 Let G be a compact Lie group of dimension d. We will be concerned with a cobordism category of smooth compact G-manifolds, with the action free on the boundary: this can be regarded as a categorical cofiber for the forgetful funct* *or from manifolds with free G-action to manifolds with unrestricted action. Under reasonable assumptions this cofiber category is closed under Cartesian products (given the diagonal action). If E is a geometric cycle theory (eg stable homotopy, or classical homology) th* *en the graded E-bordism group of free G-manifolds is isomorphic to E*+d(BG+ ). On the other hand, the homotopy quotient of a G-manifold is a bundle of manifolds over the classifying space BG, and Quillen's conventions [22] associate to such* * a thing, a class in the graded cobordism group E-*(BG+ ). The forgetful functor from free to unrestricted G-manifolds defines a long exact sequence . .!.E*-d(BG+ ) ! E-*(BG+ ) ! t-*G(E) ! E*-d-1(BG+ ) ! . . . which interprets the relative groups as the (Tate-Swan [28]) E-bordism of manif* *olds with G-action free on the boundary. The geometric boundary homomorphism @E : t-*G(E) ! E*-d-1(BG+ ) ! E*-d-1 sends a manifold with G-free boundary to the quotient of that action on the bou* *nd- ary; it will be useful later. Remarks: 1) tG (E) is a ring-spectrum if E is; in fact it is an E-algebra, at least in s* *ome naive sense; 2) tom Dieck stabilization [4] extends this geometric bordism theory to an equi* *vari- ant theory; 3) the functor tG sends cofibrations to cofibrations, but it lacks good limit p* *roper- ties: it is defined by a kind of hybrid of homology and cohomology, and Milnor's limit fails. In more modern terms [11], the construction sends a G-spectrum E to the equivariant function spectrum [EG+ , E] ^ ~EG+ . 4) The eventual focus of x2 is the case when G is the circle group T, and E is ordinary cohomology: this is closely related to cyclic cohomology [2], but I do* *n't know enough about that subject to say anything useful. 2.2 Suppose now that E is a general complex-oriented ring-spectrum; then E*(BT+* * ) is a formal power series ring generated by the Euler (or first Chern) class e. * *If E* is concentrated in even degrees, then the cofiber sequence above reduces for di* *men- sional reasons to a short exact sequence 0 ! E*(BT+ ) = E*[[e]] ! t*TE = E*((e)) ! E-*-2(BT+ ) ! 0 with middle group the ring of formal Laurent series in e. By a lemma of [12 x2.* *4] we can think of t*TE as the homotopy groups of a pro-object S2CP-11 ^ E := {S2Th (-kj) ^ E | k > -1} in the category of spectra, constructed from the filtered vector bundle . .(.k - 1)j kj (k + 1)j . . . HEISENBERG GROUPS AND ALGEBRAIC TOPOLOGY 3 defined by sums of copies of the tautological line bundle over CP 1 ~= BT, as discussed in the appendix to [6] (see also [18]). More precisely: the Thom spec* *trum can be taken to be ? Th (-kj) := limnS-2(n+1)kCPnkj , where j? is the orthogonal complement to the canonical line j in Cn+1.] When E is not complex-orientable, tTE can behave very differently: the Segal conjecture for Lie groups implies that, up to a profinite completion [10], Y tTS0 ~ S0 _ S[ BT=C] where the product runs through proper subgroups C of T (and S denotes suspen- sion). In the universal complex-oriented case, the class e-1 2 t-2TMU is represented g* *eo- metrically by the unit disk in C with the standard action of T as unit complex * *num- bers; more generally, the unit ball in Ck represents e-k. The geometric boundary homomorphism sends that T-manifold to CPk-1; this observation can be restated, using Mishchenko's logarithm, as the formula @E (f) = rese=0f(e) d logMU(e) : t*TMU ! MU -*-2 , where the algebraic residue homomorphism rese=0: MU *-2((e)) ! MU * is defined by rese=0ek de = ffik+1,0, cf. [19, 21, 29]. 2.3 The relative theory of manifolds with free group action on the boundary alo* *ne defines bordism groups ø*G(E) analogous to a truncation of Tate cohomology, with useful geometric applications. In place of the long exact sequence above, we ha* *ve . .!.E-*(S0) ! ø-*G(E) ! E*-d-1(BG+ ) ! . . . compatible with a natural transformation t*G(E) ! ø*G(E) which forgets the inte* *rior G-action. In our case (when E is complex-oriented), this is just the E-homology* * of the collapse map CP-11 ! CP-11:= Th(-j) defined by the pro-spectrum in the previous paragraph. A Riemann surface with geodesic boundary is in a natural way an orientable mani- fold with a free T-action on its boundary, and a family of such things, paramet* *rized by a space X, defines an element of ø-2TMU (X+ ) ~=[X, CP-11^ MU ] . The Hurewicz image of this element in ordinary cohomology is the homomorphism H*(CP-11, Z) ! H*(X+ , Z) defined by the classifying map of Madsen and Tillmann, which will be considered in more detail below. 4 JACK MORAVA 3. Automorphisms of classical Tate cohomology 3.1 There are profound analogies - and differences - among the Tate cohomology rings of the groups Z=2Z, T, and SU(2) [3]. A property unique to the circle is * *the existence of the nontrivial involution I : z ! z-1. When E is complex oriented, the symmetric bilinear form f, g 7! (f, g) = @E (fg) on the Laurent series ring t*TE is nondegenerate, and the involution on T defin* *es a symplectic form {f, g} = (I(f), g) which restricts to zero on the subspace of elements of positive (or negative) d* *e- gree. This Tate cohomology thus has an intrinsic inner product, with canonical polarization and involution. This bilinear form extends to a generalized Kronecker pairing tTE*(X) E tTE*(X) ! E*-2 which can be interpreted as a kind of Spanier-Whitehead duality between tTE*, viewed as a pro-object as in x2.1, and the direct system {E*(Th (-kj)) | k > -1} defined by the cohomology of that system. This colimit again defines a Laurent series ring, but this object is not quite its own dual: a shift of degree two i* *ntervenes, and it is most natural to think of the (non-existent) functional dual of tTE as S-2tTE. The residue map tTE ! S2E can thus be understood as dual to the unit ring-morphism E ! tTE. 3.2 The Tate construction is too large to be conveniently represented, so the u* *sual Hopf-algebraic approach to the study of its automorphisms is technically diffic* *ult. Fortunately, methods from the theory of Tannakian categories can be applied: we consider automorphisms of tTE as E varies, and approximate the resulting group- valued functor by representable ones. There is no difficulty in carrying this o* *ut for a general complex-oriented theory E, but the result is a straightforward extens* *ion of the case of ordinary cohomology. To start, it is clear that the group(scheme, representing the functor X A 7! G0(A) = {g(x) = gkxk+1 2 A[[x]] | g0 = 1} k 0 on commutative rings A) of automorphisms of the formal line acts as multiplicat* *ive natural transformations of the cohomology-theory-valued functor A 7! t*THA, with g 2 G0(A) sending the Euler class e to g(e). [I am treating these theories as g* *raded by Z=2Z, with A concentrated in degree zero; but one can be more careful.] Clearly G0 is represented by a polynomial Hopf algebra on generators gk, with diagonal ( g)(x) = (g 1)((1 g)(x)) . HEISENBERG GROUPS AND ALGEBRAIC TOPOLOGY 5 However, G0 is a subgroup of a larger system G of natural automorphisms, which * *is a colimit of representable functors (though not itself representable): followin* *g [16], let X p A 7! G(A) = {g(x) = gkxk+1 2 A((x)) | g0 2 Ax , gk 2 A if0 > k} k -1 be the group of invertible nil-Laurent series, ie Laurent series with g0 a unit* *, and gk nilpotent for negative k. It is clear that G is a monoid, but in fact [20] i* *t possesses inverses. The Lie algebra of G is spanned by the derivations xk+1@x, k 2 Z: it * *is the algebra of vector fields on the circle. 3.3 A related group-valued functor preserves the symplectic structure defined a* *bove: to describe it, I will specialize even further, and work over a field in whch t* *wo is invertible: R, for convenience. Thus let ~Gbe the (ind-pro)-algebraicpgroupsche* *me defined by invertible nil-Laurent series over the field R(( x)) obtained from * *R((x)) by adjoining a formalpsquare rootpof x, and let ~Godddenote the subgroup of odd invertible series ~g( x) = -~g(- x). The homomorphism p 2 ~g7! g(x) := ~g( x) : ~Godd! G is then a kind of double cover. p The functor ~Gacts by symplectic automorphisms of the module R(( x)), given the bilinear form := ß resx=0u(x) dv(x) [27]; it is in fact a grouppof restricted symplectic automorphisms of this modu* *le. The Galois group of R(( x))=R((x)) defines a Z=2Z - action, and the subgroup ~Goddpreserves the subspace R((p x))oddof odd power series. Proposition The linear transformation p t*THR ! R(( x))odd defined on normalized basis elements by ek 7! fl-k-1_(x) 2 (where fls(x) = (1 + s)-1xs denotes a divided power), is a dense symplectic em- bedding. Proof: We have {ek, el} = (-1)k rese=0ek+lde = (-1)kffik+l+1,0 while = resx=0fls(x)flt-1(x) dx = _____ß_____f(t)f(1i+ss)+t,0. The assertion then follows from the duplication formula for the Gamma function. p The half-integral divided powers lie in Q(( x)), aside from distracting powers* * of ß. The remaining rational coefficients involve the characteristic `odd' factori* *als of 2D topological gravity [8, 15], eg when k is positive, p (k + 1_2) = (2k - 1)!! 2-k ß . 6 JACK MORAVA 4.Symmetries of the stable cohomology of the Riemann moduli space The preceding sections describe the construction of a polarized symplectic stru* *cture on the Tate cohomology of the circle group. The algebra of symmetric functions on the Lagrangian submodule H*(CP+1, Q) tTHQ of that cohomology can be identified with the homology of the infinite loopspace QCP+1 = limn nSn CP+1 ; on the other hand, this module of functions admits a canonical action of the He* *isen- berg group associated to its defining symplectic module [24 x9.5]. The point of this paper is that the homology of this infinite loopspace, consid* *ered in this way as a Fock representation, manifests the Virasoro representation constr* *ucted by Witten and Kontsevich on the stable cohomology of the moduli space of Riemann surfaces, identified with H*(QCP+1, Q) through work of Madsen, Tillman, and Weiss. Some of those results are summarized in the next two subsections; a more thorough account can be found in Michael Weiss's survey in these Proceedings. T* *he third section below discusses their connection with representation theory. 4.1 Here is a very condensed account of one component of [17]: if F Rn is a c* *losed connected two-manifold embedded smoothly in a high-dimensional Euclidean space, its Pontrjagin-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 classified 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 pul* *lback along ø of (n - j); composing the map induced on Thom spaces with the collapse defines the map 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. This construction factors through a map a BDiff(Fg) ! Z x B +1! 1 CP-11 g 0 of infinite loopspaces. Collapsing the bottom cell defines a cofibration S-2 ! CP-11! CP+1 of spectra; the fiber of the corresponding map 2QS0 ! 1 CP-11! QCP+1 HEISENBERG GROUPS AND ALGEBRAIC TOPOLOGY 7 of spaces has torsion homology, and the resulting composition Z x B +1! QCP+1 ~ QS0 x QCP 1 is a rational homology isomorphism which identifies Mumford's polynomial algebra on classes ~i, i 1, with the symmetric algebra on positive powers of e. The rational cohomology of QS0 adds a copy of the group ring of Z, which can be interpreted as a ring of Laurent series in a zeroth Mumford class ~0. The standard convention is to write bk for the generators of H*CP+1 dual to ek,* * and to use the same symbols for their images in the symmetric algebra H*(QCP+1, Q). The Thom construction defines a map CP 1 ! MU which extends to a ring isomorphism H*(QCP 1, Q) ! H*(MU , Q) ; sending the bk to classes usually denoted tk, with k 1; but it is convenient * *to extend this to allow k = 0. 4.2 The homomorphism limMU *+n-2(Th (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, the Thom class of n - j is its * *zero- section, regarded as a map between manifolds. Its image is the class defined by* * the fiber product Zn _____________//Grass2,n | | | | fflffl| fflffl| Rn+^DiffEmb(F )_____//Grass(n-j)2,n; this is the space of equivalence classes, under the action of Diff(F ), of pair* *s (x, OE), with x 2 OE(F ) Rn a point of the surface (ie, in the zero-section of ), and* * OE an embedding. Up to suspension, this image is thus the element [Zn ! Rn+^DiffEmb] 7! MU n-2(SnBDiff(F )) defined by the tautological family F xDiffEDiff(F ) of surfaces over the classi* *fying space of the diffeomorphism group. It is primitive in the Hopf-like structure d* *efined by gluing: in fact it is the image of X ~ktk+1 2 MU -2(B +1) Q . k 1 If v is a formal indeterminate of cohomological degree two, then the class = exp(vTh (-j)) 2 MU 0Q( 1 CP-11)[[v]] defined by finite unordered configurations of points on the universal surface (* *with v a book-keeping indeterminate of cohomological degree two) is a kind of exponent* *ial transformation ~ *: H*(QCP+1, Q) ! H*(MU , Q[[v]]) . 8 JACK MORAVA From this perspectivePit is natural to interpret the Thom class in MU -2(CP-11)* * Q as the sum k -1tk+1ek, with t0 = v-1e. 4.3 A class in the cohomology group H2Lie(V, R) ~= 2(V *) of a real vector space V defines a Heisenberg extension 0 ! T ! H ! V ! 0 ; the representation theory of such groups, and in particular the construction of their Fock representations, is classical [5]. What is important to us is that t* *hese are projective representations of V , with positive energy; such representations ha* *ve very special properties. The loop group of a circle is a key example; it possesses an intrinsic symplect* *ic form, defined by formulae much like those of x2 [23 x5, x7b]. Diffeomorphisms of the circle act on any such loop group, and it is a deep property of positive-en* *ergy representations, that they extend to representations of the resulting semidirect product of the loop group by DiffS1. Therefore by restriction a positive-energy representation of a loop group automatically provides a representation of DiffS* *1. This [Segal-Sugawara [25 x13.4]] construction yields the action of Witten's Vir* *asoro algebra on the Fock space Symm (H*(CP+1)) ~=Q[tk | k 0] . In Kontsevich's model, the classes tk are identified with the symmetric functio* *ns Traceflk-1_( 2) ~ -(2k - 1)!! Trace -2k-1 2 of a positive-definite Hermitian matrix . Note, however, that the deeper results of Kontsevich and Witten theory [31] are inaccessible in_this toy model: that theory is formulated in terms`of_compactif* *ied moduli spaces M gof algebraic curves. The rational homology of Q( M g) (suitab* *ly interpreted, for small g) contains a fundamental class X __ 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 killed b* *y the subalgebra of Virasoro generated by the operators Lk with k -1. 5. Concluding remarks 5.1 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 [14]. The group of automorphisms of suc* *h 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 Th(-jr), and it is reasonable to expect that this HEISENBERG GROUPS AND ALGEBRAIC TOPOLOGY 9 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 Th (-jp) ! Th(-jup) of spectra, and the classification of fiber-homotopy equivalences of vector bun* *dles yields an equivalence of Th(-jup) with Th(-jp) after p-completion. There is an analogous decomposition of tTHZp and a corresponding decomposition of the as- sociated Fock representations [20 x2.4]. 5.2 Tillmann has also studied categories of surfaces above a parameter space X; the resulting group completions have interesting connections with both Tate and quantum cohomology. When X is a compact smooth almost-complex manifold, its Hodge-deRham cohomology admits a natural action of the Lie algebra generated by the Hodge dimension operator H together with multiplication by the first Che* *rn class (E) and its adjoin (F = *E*) [26]. Recently Givental [9 x8.1] has shown t* *hat earlier work of (the schools of) Eguchi, Dubrovin, and others can be reformulat* *ed in terms of structures on t*,*THdg(X), given a symplectic structure generalizin* *g that of x3. In this work, the relevant involution is IGiv= exp(1_2H) exp(-E) I exp(E) exp(-1_2H) ; it would be very interesting if this involution could be understood in terms of* * the equivariant geometry of the free loopspace of X [7]. 5.3 Nothing forces us to restrict the construction of Madsen and Tillmann to tw* *o- manifolds, and I want to close with a remark about the cobordism category of smooth spin four-manifolds bounded by ordinary three-spheres. A parametrized family of such objects defines, as in x2.3, an element of the truncated equivar* *iant cobordism group ø-4SU(2)MSpin(X+ ) . On the other hand, it is a basic fact of four-dimensional life that Spin(4) = SU(2) x SU(2) , so the Madsen-Tillmann spectrum for the cobordism category of such spin four-fo* *lds is the twisted desuspension * V BSpin(4)-æ = (HP1 x HP1 )-V H of the classifying space of the spinor group by the representation æ defined by the tensor product of two standard rank one quaternionic modules over SU(2) [13 x1.4]. Composition with the Dirac operator defines an interesting rational homo* *logy isomorphism * V -V -V (HP1 x HP1 )-V H ! HP1 ^ MSpin ! HP1 ^ kO related in low dimensions to the classification of unimodular even indefinite l* *attices [27, 30]. This suggests that the Tate cohomology t*SU(2)kOmay have an interesti* *ng role to play in the study of topological gravity in dimension four. 10 JACK MORAVA References 1. J.F. Adams, . .w.hat we don't know about RP1 , in New Developments in Topolo* *gy, ed. G. Segal, LMS Lecture Notes 11 (1972) 1 - 9 2. A. Adem, R.L. Cohen, W. Dwyer,Generalized Tate homology, homotopy fixed poin* *ts and the transfer, in Algebraic topology (Evanston 88) 1 - 13, Contemp. Math. 96 (198* *9) 3. V.I. Arnol'd, Symplectization, complexification and mathematical trinities, * *in The Arnold- fest 23 - 37, Fields Inst. Commun. 24 (1999) 4. Th. Bröcker, E.C. Hook, Stable equivariant bordism, Math. Zeits. 129 (1972) * *269 - 277 5. P. Cartier, Quantum mechanical commutation relations and theta functions, in* * Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math. 9 (1966) 361 - * *383 6. 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 7. __-, A. Stacey, Fourier decompositions of loop bundles, Proc. Northwestern C* *onf (2001), to appear 8. P. DiFrancesco, C. Itzykson, J.-B. Zuber, Polynomial averages in the Kontsev* *ich model, CMP 151 (1993) 193-219 9. A. Givental, Gromov - Witten invariants and quantization of quadratic hamilt* *onians, available at math.AG/0108100 10.J.P.C. Greenlees, A rational splitting theorem for the universal space for a* *lmost free actions, Bull. London Math. Soc. 28 (1996) 183 - 189 11.__-, J.P. May, Generalized Tate cohomology, Mem. AMS 113 (1995) 12.__-, H. Sadofsky, The Tate spectrum of vn-periodic complex-oriented theories* *, Math. Zeits. 222 (1996) 391 - 405 13.R. Gompf, A. Stipsicz Four-manifolds and Kirby calculus, AMS Grad Texts 20 (* *1999) 14.T. Jarvis, T. Kimura, A. Vaintrob, Moduli spaces of higher spin curves and i* *ntegrable hierar- chies, available at math.AG/9905034 15.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 16.M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space, avai* *lable at math.AG/0107143 17.I. Madsen, U. Tillmann, The stable mapping-class group and Q(CP1+), Invent. * *Math. 145 (2001) 509 - 544. 18.M. Mahowald, On the metastable homotopy of Sn, Mem. AMS 72 (1967) 19.J. Morava, Cobordism of involutions revisited, revisited, in The Boardman Fe* *stschrift, Contemporary Math. 239 (1999) 20.__-, An algebraic analog of the Virasoro group, Czech. J. Phys. 51 (2001), a* *vailable at math.QA/0109084 21.D. Quillen, On the formal group laws of unoriented and complex cobordism the* *ory, BAMS 75 (1969) 1293 - 1298 22.__-, Elementary proofs of some results of cobordism theory using Steenrod op* *erations, Adv. in Math 7 (1971) 29 - 56 23.G. Segal, Unitary representations of some infinite-dimensional groups, Comm.* * Math. Phys. 80 (1981) 301 - 342. 24.__-, A. Pressley, Loop groups, Oxford (1986) 25.__-, Algebres de Lie semisimples complexes, Benjamin (1966) 26.J.P. Serre, Corps Locaux, Hermann (1968) 27.__-, A Course in Arithmetic, Springer (1973) 28.R. Swan, Periodic resolutions for finite groups, Annals of Math. 72 (1960) 2* *67 - 291 29.J. Tate, Residues of differentials on curves, Ann. Sci. Ecole Norm. Sup. 1 (* *1968) 149 - 159 30.C.T.C. Wall, On simply-connected four-manifolds, J. London Math. Soc. 39 (19* *64) 141 - 149 31.E. Witten, Two-dimensional gravity and intersection theory on moduli space, * *Surveys in Differential Geometry 1 (1991) 243 - 310 Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218 E-mail address: jack@math.jhu.edu