Tautological classes of definite 4-manifolds

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G eometry &

T opology

msp

Volume 27 (2023)

Tautological classes of definite 4–manifolds

DAVID BARAGLIA

Published: 16 May 2023

Tautological classes of definite 4–manifolds

DAVIDBARAGLIA

We prove a diagonalisation theorem for the tautological, or generalised Miller–Morita–

Mumford, classes of compact, smooth, simply connected, definite4–manifolds. Our result can be thought of as a families version of Donaldson’s diagonalisation theorem.

We prove our result using a families version of the Bauer–Furuta cohomotopy refine- ment of Seiberg–Witten theory. We use our main result to deduce various results concerning the tautological classes of such4–manifolds. In particular, we completely determine the tautological rings ofCP²andCP²#CP². We also derive a series of linear relations in the tautological ring which are universal in the sense that they hold for all compact, smooth, simply connected definite4–manifolds.

53C07, 57K41, 57R22

1. Introduction 642

2. Families of definite4–manifolds 649

3. Families Bauer–Furuta theory 654

4. Cohomology rings of families 660

5. Tautological classes 670

6. CP² 689

7. CP²#CP² 693

8. Linear relations in the tautological ring 695

List of symbols 696

References 697

© 2023 MSP (Mathematical Sciences Publishers). Distributed under theCreative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions viaSubscribe to Open.

1 Introduction

1.1 Tautological classes

Let X be a compact, simply connected smooth 4–manifold with positive-definite intersection form. Assume thatb2.X / > 0. Then, by the work of Donaldson[5]and Freedman[8],X is homeomorphic to the connected sum

#

^{nCP2 ofn}1copies ofCP², wherenDb2.X /.

LetWE!B be a compact, smooth family with fibres diffeomorphic toX. By this we mean thatE and B are compact, smooth manifolds, is a proper submersion and each fibre of with its induced smooth structure is diffeomorphic toX. Note thatE has a fibrewise orientation which is uniquely determined by the requirement that the fibres of E are positive-definite 4–manifolds. In this paper, we will use parametrised Seiberg–Witten theory to study thetautological classes, orgeneralised Miller–Morita–Mumford classes, of such families. These are defined as follows. Let T .E=B/DKer.WTE!TB/denote the vertical tangent bundle. Then, for each rational characteristic classc2H.BSO.4/IQ/, we define the associated tautological class as

c.E/D ˆ

E=B

c.T .E=B//2H ⁴.BIQ/;

where ´

E=B denotes integration over the fibres. Let Diff.X / denote the group of diffeomorphisms of X with the _C¹–topology (note that all diffeomorphisms of X are orientation-preserving sinceX is positive-definite) andBDiff.X /the classifying space. The tautological classes can be constructed for the universal bundle UX D EDiff.X /Diff.X /X, giving classes

cDc.UX/2H.BDiff.X /IQ/:

Thetautological ringofX,

R.X /H.BDiff.X /IQ/;

is defined as the subring of H.BDiff.X /IQ/generated by tautological classesc

forc2H.BSO.4/IQ/. SinceH.BSO.4/IQ/is generated overQbyp1 ande, it follows thatR.X /is generated by the classesf_p^a

1e^bga;b0. Similarly, for any family E!B, we can define the tautological ring ofE,

R.E/H.BIQ/;

to be the subring generated by the tautological classesc.E/forc2H.BSO.4/IQ/.

Tautological rings have been studied extensively for families of oriented surfaces, eg by Mumford[18], Miller[15], Looijenga[14], Faber[7] and Morita[16], and there is a growing literature on tautological classes in higher dimensions due to Galatius, Grigoriev, Hebestreit, Land, Lück and Randal-Williams[10;11;9;19;13]and Busta- mante, Farrell and Jiang[4]. However, as far as we are aware, our paper is the first to use gauge theory to obtain results on the tautological classes of4–manifolds.

LetBbe a compact smooth manifold. A topological fibre bundleE!Bwith transition functions valued in Diff.X / may be obtained by pullback of the universal family UX !BDiff.X /with respect to a continuous mapB!BDiff.X /. As explained by Baraglia and Konno[2, Section 4.2], it follows from a result of Müller and Wockel[17]

that such a familyE!Badmits a smooth structure for which is a submersion and the fibres ofE with their induced smooth structure are diffeomorphic toX. SinceE is smooth, we may use parametrised gauge theory to study the tautological classes c.E/2H.BIQ/. If a relation amongst tautological classes holds inR.E/for all compact, smooth familiesWE!B with fibres diffeomorphic toX, then it must also hold inR.X /. This is because rational cohomology classes ofBDiff.X /are detected by continuous maps from compact, smooth manifolds intoBDiff.X /. The upshot of this is that we can use gauge theory to indirectly study the tautological ring ofX.

1.2 Main results

In our first main result we determine the tautological rings ofCP²andCP²#CP². Theorem 1.1 The tautological rings of CP²andCP²#CP²are given by:

(1) R.CP²/DQŒ_p²

1; _p⁴

1.

(2) R.CP²#CP²/DQŒ_p2 1; _p3

1.

VariantsR.X;/andR.X; D⁴/of the tautological ring are defined in[11;9]. Their definition is recalled inSection 6. We determine these rings forCP².

Theorem 1.2 We have ring isomorphisms: (1) R.CP²;/ŠQŒp1; e.

(2) R.CP²; D⁴/ŠQ.

The ringsR.CP²/, R.CP²;/andR.CP²; D⁴/were investigated in [19], but their structure was not fully determined. By computing these rings we have settled some open problems posed in[19].

For each pair of nonnegative integersaandb, we define a two-variable polynomial a;b.x; y/2ZŒx; yas follows: Let

p.z/Dz³ xz y; p⁰.z/D3z² x:

Then we define

a;b.x; y/D 1 2 i

‰ .p⁰.z/C3x/^a.p⁰.z//^b

p.z/ dz;

where the contour encloses all zeros ofp.z/. From this definition, it follows thata;b

satisfies the recursive formulas

_aC1;b.x; y/D_a;bC1.x; y/C3x_a;b.x; y/;

_a;bC3.x; y/D3x_a;bC2.x; y/C.27y² 4x³/_a;b.x; y/;

which, together with the initial conditions

0;0.x; y/D0; 0;1.x; y/D3; 0;2.x; y/D3x;

can be used to compute _a;b for all values ofa and b. We will make use of the polynomials_a;b in the computation of tautological classes of families of definite 4–manifolds. We first state thenD1case.

Theorem 1.3 LetE!Bbe a smooth family with fibres diffeomorphic toX,whereX is a smooth,compact,simply connected,positive-definite4–manifold withb2.X /D1.

Suppose that the monodromy action of 1.B/onH².XIZ/is trivial. Then there exist classesB2H⁴.BIQ/,C 2H⁶.BIQ/such that:

(i) There is an isomorphism ofH.BIQ/–algebras

H.EIQ/ŠH.BIQ/Œx=.x³ Bx C /:

(ii) The Euler class and first Pontryagin class ofT .E=B/are given by eD3x² B; p1D3x²C2B:

(iii) For all a; b0,

_p^a

1e^b.E/D_a;b.B; C /:

An interesting consequence ofTheorem 1.3is that the rational cohomology classp1

depends only on the underlying topological structure of the family becausep1is com- pletely determined byxandBand in turn these classes can be uniquely characterised in terms of the pushforward mapWH.EIQ/!H ⁴.BIQ/. In fact, a similar

result is true more generally for families of definite4–manifolds. SeeRemark 5.12. It follows that the tautological classes_p^a

1e^b.E/depend only on the underlying topological structure of the family. This could also be deduced from the fact that the tautological classes are also defined for topological bundles; see Ebert and Randal-Williams [6, Theorem B].

Remark 1.4 IfE DP.V /is theCP²–bundle associated to a complex rank3vec- tor bundle V !B with trivial determinant, then B D c2.V / and C D c3.V /, and so _p^a

1e^b.E/D_a;b. c2.V /; c3.V //, which gives the tautological classes as polynomials inc2.V /andc3.V /.

To state our next result, we need a few definitions. Letƒndenote a free abelian group of ranknand letfe1; : : : ; engbe a basis. Equipƒnwith the standard Euclidean inner producthei; eji Dıij. LetWndenote the isometry group ofƒn. ThenWnis isomorphic to a semidirect product

WnDSnËZⁿ₂;

where the symmetric groupSnacts by permutation — .ei/De _{.i /}— and the normal subgroupZⁿ₂is generated by1; : : : ; n, wherei is the reflection in the hyperplane orthogonal toei. LetX denote a smooth, compact, simply connected4–manifold with positive-definite intersection form and nDb2.X /1. InSection 2 we construct a principalWn–bundle

pWBDiff.X /!BDiff.X /

overBDiff.X /. Sincepis a finite covering, the pullback mappWH.BDiff.X /IQ/! H.BDiff.X /IQ/ is injective and the image is precisely the Wn–invariant part of H.BDiff.X /IQ/. In particular, we may think of the tautological ringR.X / as a subring ofH.BDiff.X /IQ/.

Theorem 1.5 Let X be a smooth, compact, simply connected, positive-definite 4–manifold withb2.X /Dn2. Then there exists classes

Dij 2H².BDiff.X /IQ/; 1i; j n; i¤j with the following properties:

(i) LetD.X /H.BDiff.X /IQ/be the subring generated byfDi;jgi¤j. The groupWnacts on the subringD.X /according to

.Dij/DD _{.i / .j /} for 2Sn and _k.Dij/D

Dij if k¤j;

Dij if kDj:

(ii) LetI.X /denote theWn–invariant subring ofD.X /. Then I.X /H.BDiff.X /IQ/:

(iii) The tautological ringR.X /of Xis a subring ofI.X /. That is,all tautological classes of X can be expressed asWn–invariant polynomials of theDij.

Theorem 1.5says that the tautological classes can be written asWn–invariant polyno- mials of theDij. The next theorem addresses the question of how to compute these invariant polynomials. First we set

I1DX

i;j i¤j

D²_ij; I2D X

i;j;k i;j;kdistinct

D_{i k}D_{j k}:

Then, fori D1; : : : ; n, define BiD3

2 X

j j¤i

D²_ij n 5

2n.n 1/I1 1

n 1I2; CiD 1 n 1

X

j j¤i

.D_{j i}³ BiDj i/:

Theorem 1.6 Let X be a smooth, compact, simply connected, positive-definite 4–manifold withb2.X /Dn2. Then:

(1) The tautological classesc withcDp₁^ae^2b are given by _p^a

1e^2b D

n

X

iD1

a;2b.Bi; Ci/:

(2) The tautological classesc withcDp₁^ae^2bC1are given by _p^a

1e^2bC1D

n

X

iD1

a;2bC1.Bi; Ci/ 2X

j j¤i

.3D²_ijC2Bj/^a.3D_ij² Bj/^2b:

Theorem 1.6gives a completely explicit expression for the tautological classesc as polynomials infDijgi¤j once the polynomials_a;b.x; y/are known. As an application ofTheorem 1.6, we prove the existence of many linear relations amongst tautological classes.

Theorem 1.7 Let X be a smooth,compact,simply connected,definite4–manifold and let d1be given. Then,amongst all tautological classes_p^a

1e^b withaCbDd and beven,there are at least

₁

2d˘ ₁

3.d 1/˘

linear relations. More precisely,if c0; c1; : : : ; c_bd=2c2Qare such that (1-1)

bd=2c

X

jD0

cjd 2j;2j.x; y/D0;

then we also have

bd=2c

X

jD0

cj_pd 2j

1 e^2j D0

and the space of .c0; c1; : : : ; c_bd=2c/satisfying(1-1)has dimension at least ₁

2d˘ ₁

3.d 1/˘ :

As explained inSection 8, for eachd2, the families signature theorem gives one linear relation amongst the tautological classes_p^a

1e^b withaCbDd andbeven.Theorem 1.7 implies that there are further linear relations whenever₁

2d˘ ₁

3.d 1/˘

> 1. This is the case ifd D6ord 8. The first few such relations (up todD12) are

0D4_p4

1e² 41_p2

1e⁴C100_e6; 0D36_p6

1e² 461_p4

1e⁴C1843_p2

1e⁶ 2300_e8; 0D24_p7

1e² 322_p5

1e⁴C1379_p3

1e⁶ 1900_p1e8; 0D108_p8

1e² 1579_p6

1e⁴C7902_p4

1e⁶ 15 531_p2

1e⁸C9100_e10; 0D360_p9

1e² 5606_p7

1e⁴C30 923_p5

1e⁶ 71 311_p3

1e⁸C57 100_p1e10; 0D144_p8

1e⁴ 2552_p6

1e⁶C16 629_p4

1e⁸ 47 400_p2

1e¹⁰C50 000_e12; 0D6000_p10

1 e² 98 012_p8

1e⁴C577 796_p6

1e⁶ 1 461 667_p4

1e⁸C1 338 700_p2 1e¹⁰: 1.3 Idea behind main results

The inspiration for our main results comes from considering Donaldson theory for a family of definite 4–manifolds. Let X be a compact, simply connected smooth 4–manifold with positive-definite intersection form. Recall the proof of Donaldson’s diagonalisation theorem uses the moduli space_Mof selfdual instantons on an SU.2/–

bundleE!X withc2.E/D 1. Then_Mis a5–dimensional oriented manifold with singularities. The singularities correspond to reducible instantons, which correspond to elements in 2H².XIZ/ satisfying²D1, considered modulo 7! . Each singularity of _Mtakes the form of a cone over CP². The moduli space_Mis non- compact, but it admits a compactification_Mwhose boundary is diffeomorphic toX.

Removing from_Ma neighbourhood of each singularity, we obtain a cobordism_M⁰ fromX to a disjoint union of copies ofCP². Cobordism-invariance of the signature implies that there arenDb2.X /copies ofCP² and, hence, there arendistinct pairs of elements˙1; : : : ;˙n2H².XIZ/satisfying_i²D1. This implies thatH².XIZ/

is diagonalisable.

Now suppose that E !B is a smooth family with fibres diffeomorphic to X and suppose for simplicity that the monodromy action of1.B/onH².XIZ/is trivial.

Considering the moduli space of selfdual instantons withc2.E/D 1on each fibre ofE, we obtain a families moduli space_ME !B. Note that_ME is typically not a fibre bundle since the topology of the fibres of_ME can vary as we move inB. We would expect that, for a sufficiently generic family of metrics onE, we can arrange that ME is smooth away from reducible solutions and that the structure of_ME around the reducibles is given by taking fibrewise cones onnCP²–bundlesE1; : : : ; EnoverB.

We would further expect that_ME can be compactified by adding a boundary which is diffeomorphic to the familyE. Removing a neighbourhood of the reducible solutions, we would expect to obtain a cobordism⁰WM⁰_E !BrelativeB, betweenE!B and the disjoint union ofCP²–bundlesE1; : : : ; En. Consider the virtual vector bundle V DT_M⁰_E .⁰/.TB/. ClearlyVj^E DT .E=B/andVj^Ei DT .Ei=B/for eachi; hence, the Pontryagin classes ofV restrict to the Pontryagin classes ofEandEi on the boundary. Applying Stokes’ theorem, we expect to obtain a kind of “diagonalisation theorem” for the tautological classes:

(1-2) _p^a

1e^2b.E/D

n

X

iD1

_p^a

1e^2b.Ei/:

Note that we need to take even powers ofebecausee is unstable wherease²Dp2is stable.

There are some technical challenges for carrying out this argument rigorously. Most notably, it seems difficult to arrange unobstructedness of the families moduli space around the reducible solutions. It is well known that this can be done for a single moduli space by choosing a sufficiently generic metric, but extending this to families appears challenging. Nevertheless, the intuition provided by Donaldson theory turns out to be essentially correct. Theorem 1.6provides a rigorous version of(1-2), where a;b.Bi; Ci/plays the role of_p^a

1;e^b.Ei/. Note, byRemark 1.4, that, if Bi and Ci

are the Chern classes of a rank3vector bundleVi!Bwith trivial determinant, then _a;b.Bi; Ci/D_p^a

1e^b.Ei/, whereEi is theCP²–bundle Ei DP.Vi/. The natural

candidate for Vi is the families index of the instanton deformation complex around the corresponding reducible, except that we only know this exists as avirtualvector bundle.

We will prove the main results using families Seiberg–Witten theory, or, more pre- cisely, the Bauer–Furuta cohomotopy refinement of Seiberg–Witten theory. The main advantage of this approach is that it allows us to avoid various transversality issues that typically arise in the construction of moduli spaces. It is quite surprising that Seiberg–Witten theory works here. The issue is that the families Seiberg–Witten moduli space is compact and there is no obvious relation between the families moduli space and the familyE. What happens instead is that Seiberg–Witten theory gives constraints on the topology of the families index associated to families of Dirac operators onE. We use this to indirectly obtain a series of constraints on the cohomology ringH.EIQ/

of the familyE and in turn this gives constraints on the tautological classes.

Outline of the paper

InSection 2, we establish some basic results concerning families of definite4–manifolds.

InSection 3, we consider the Bauer–Furuta refinement of Seiberg–Witten theory for a family of definite4–manifolds. The main result isTheorem 3.1. The rest of the section is concerned with understanding some of the implications of this theorem. InSection 4, we study in great detail the structure of the cohomology ringsH.EIQ/of families of definite 4–manifolds and inSection 5we prove our main results concerning the tautological classes of such families. Sections6and7are concerned with the special cases ofCP²andCP²#CP²and finally, inSection 8, we study linear relations in the tautological rings of definite4–manifolds.

Acknowledgements We thank Oscar Randal-Williams for helpful comments on the paper and for suggesting a simpler proof ofLemma 3.2. The author was financially supported by the Australian Research Council Discovery Project DP170101054.

2 Families of definite 4–manifolds

Throughout,X denotes a smooth, compact, simply connected4–manifold with positive- definite intersection form andnDb2.X /1. The intersection pairingh;ionH².X / is a symmetric, unimodular bilinear form. By Donaldson’s diagonalisation theorem[5],

the intersection form on H².XIZ/ is diagonal and so there exists an orthonormal basisf1; : : : ; ngforH².XIZ/. An orthonormal basis forH².XIZ/will be called aframingofH².XIZ/. Letƒndenote the free abelian groupZⁿof ranknand let fe1; : : : ; engbe the standard basis. Equipƒnwith the standard Euclidean inner product.

Then a framingf1; : : : ; ngofH².XIZ/determines an isometryWƒn!H².XIZ/

given by.ei/Di. LetWnDAut.ƒn/denote the symmetry group ofƒnequipped with its intersection form. Since the only classes of norm1are ˙e1; : : : ;˙en, it is easy to see thatWnis isomorphic to a semidirect product

WnDSnËZⁿ₂;

where the symmetric groupSnacts by permutation — .ei/De .i /— and the normal subgroupZⁿ₂is generated byf1; : : : ; ng, wherei is the reflection in the hyperplane orthogonal toei:

i.ej/D

ej ifj ¤i;

ej ifj Di:

Wnis also the Weyl group of the root systemsBnandCn. Clearly,Wnis a subgroup of the isometry group ofƒn. For the reverse inclusion, note that any isometry must permute the vectors of unit length, which are˙e₁;˙e₂; : : : ;˙e_n. Hence, any isometry ofƒ_n is given by a permutation offe1; : : : ; eng, followed by some sign changesei 7! ei. Let Diff.X /denote the group of orientation-preserving diffeomorphisms ofX with the C¹–topology and Diff0.X /the subgroup acting trivially onH².XIZ/. Equivalently, Diff0.X /is the subgroup of Diff.X /preserving a framing ofH².XIZ/. By definition, we have a short exact sequence

1!Diff0.X /!Diff.X /!K.X /!1;

where K.X / is the image of the map Diff.X / ! Aut.H².XIZ// which sends a diffeomorphismf WX!Xto the induced map.f ¹/WH².XIZ/!H².XIZ/. Note that, ifXD

#

^{nCP2, thenK.X /}DAut.H².XIZ/. One sees this as follows: There is an orientation-preserving diffeomorphism ofCP²which acts as 1onH².CP²IZ/, namely complex conjugation. Such a diffeomorphism can be isotoped so as to act as the identity on a disc inCP², and hence can be extended to the connected sum

#

^nCP2.

Since we can do this for each summand, we see that1; : : : ; n2K.X /. To see that SnK.X /, regardX asS⁴ withCP²attached atnpoints. Since thesenpoints can be permuted by diffeomorphisms ofS⁴, it follows thatSnK.X /.

Fixing a framing1; : : : ; nofH².XIZ/, we can identifyK.X /with a subgroup ofWn. SinceWnis finite, so isK.X /. Taking classifying spaces, we see thatBDiff0.X /has

the structure of a principalK.X /–bundle overBDiff.X /. We now define BDiff.X /DBDiff0.X /K.X /Wn:

SoBDiff.X /is a principalWn–bundle overBDiff.X /. LetpWBDiff.X /!BDiff.X / be the covering map. Since p is a finite covering, it follows that the pullback map pWH.BDiff.X /IQ/!H.BDiff.X /IQ/ is injective and that the image is pre- cisely theWn–invariant part ofH.BDiff.X /IQ/. Therefore, we may identify the tautological ringR.X /with a subring ofH.BDiff.X /IQ/:

R.X /H.BDiff.X /IQ/H.BDiff.X /IQ/:

Remark 2.1 SinceBDiff.X /DBDiff0.X /K.X /Wn, we have a fibration BDiff0.X /!BDiff.X /!Wn=K.X /:

But Wn=K.X / is a finite discrete set, so BDiff.X / is just the disjoint union of jWn=K.X /jcopies ofBDiff0.X /. For this reason it makes little difference whether we work withBDiff0.X /orBDiff.X /. We prefer to useBDiff.X /because the whole isometry group Wn acts on this space. Note also that, for X D

#

^{nCP2, we have}

K.X /DWnand soBDiff.X /DBDiff0.X /in this case.

LetWE!Bbe a family with fibres diffeomorphic toX. ThenE admits a reduction of structure to Diff0.X /if and only if the monodromy action of1.B/onH²of the fibres is trivial. In such a case, if we choose a framingf1; : : : ; ngof a single fibre and parallel translate, we obtain a framingf1.b/; : : : ; n.b/gofH².X_bIZ/for eachb2B such that the framing varies continuously withb. Henceforth we will restrict attention to familiesWE!Bequipped with a reduction of structure group to Diff0.X /. We assume further that a framing has been chosen.

Proposition 2.2 Let WE!Bbe a family with structure groupDiff0.X /. Then the Leray–Serre spectral sequence forH.EIQ/degenerates atE2.

Proof It suffices to prove the result whenBis connected. Lete2H⁴.EIQ/denote the Euler class of the vertical tangent bundle. For eachb 2B, we have thatej^Xb is 2Cntimes a generator ofH⁴.XbIQ/. It follows that all the differentials of the form drWE_r^0;4!E_r^{r;5 r} are zero. Moreover, the differentials forr odd are all zero because H.XIQ/is nonzero only in even degrees. Next, note thatE₂^0;2ŠH².XIQ/(since

Bis connected). Thus, we can identify1; : : : ; nwith classes inE₂^0;2. Now_j²2E₂^0;4, so

0Dd3._j²/D2jd3.j/2E₃^3;2ŠH².XIQ/˝H³.BIQ/:

Hence,d3.j/D0. It follows that there exist classesx1; : : : xn2H².EIQ/such that xjj^Xb Dj.b/. Now the result follows by the Leray–Hirsch theorem.

As seen in the proof of Proposition 2.2, there exist classes x1; : : : xn 2H².EIQ/

such thatxjjXb Dj.b/(note that in general the classes xj can’t be taken to lie in H².EIZ/). Thexi are not unique because, ifa2H².BIQ/, thenxjC.a/also restricts toj.b/onX_b. From the Leray–Serre spectral sequence it is clear that thexj

are unique up to such shifts.

Let ˆ

E=BWH^k.EIQ/!H^{k 4}.BIQ/

denote fibre integration. We clearly have ˆ

E=B

xj D0;

ˆ

E=B

x_j²D1;

ˆ

E=B

xixj D0

for alliandj withj¤i. Lete2H⁴.EIQ/denote the Euler class andpj2H^4j.EIQ/

the Pontryagin classes of the vertical tangent bundle. Since the fibres are4–dimensional, we havepj D0 forj > 2andp2De². So all rational characteristic classes of the vertical tangent bundle can be expressed in terms ofp1ande. From the Gauss–Bonnet and signature theorems, we have

ˆ

E=B

eD.X /DnC2;

ˆ

E=B

p1D3 .X /D3n:

Proposition 2.3 LetWE!Bbe a family with structure groupDiff0.X /and framing 1; : : : ; n. Then there exist uniquely determined classesx1; : : : xn2H².EIQ/and 2H⁴.EIQ/such that:

(1) xjj^Xb Dj.b/forj D1; : : : n.

(2) ´

E=Bx_j³D0forj D1; : : : n.

(3) ´

E=BD1.

(4) ´

E=BxjD0forj D1; : : : n.

(5) ´

E=B²D0.

Proof We already saw that there exist classes y1; : : : ; yn 2 H².EIQ/ such that yjjXb Dj.b/. Now set

xj Dyj 1 3

ˆ

E=B

y_j³

:

Then it is straightforward that thexj satisfy (1) and (2). Now let0Dx₁²2H⁴.EIQ/.

This satisfies (3). Now set 1D0

n

X

jD1

ˆ

E=B

xj0

xj:

Then1 clearly satisfies (3) and (4). Moreover, any other class satisfying (3) and (4) must be of the form1C.a/for somea2H⁴.BIQ/. Set

D1 1 2

ˆ

E=B

₁²

:

Thensatisfies (3)–(5). Uniqueness of and thexi is straightforward.

In summary,H.EIQ/is a freeH.BIQ/–module with a uniquely determined basis 1; x1; : : : ; xn; satisfying:

(1) ´

E=B1D0.

(2) ´

E=Bxj D0forj D1; : : : n.

(3) ´

E=Bx_j²D1forj D1; : : : n.

(4) ´

E=Bxixj D0fori; j D1; : : : nwithi ¤j.

(5) ´

E=Bx_j³D0forj D1; : : : n.

(6) ´

E=BD1.

(7) ´

E=BxjD0forj D1; : : : n.

(8) ´

E=B²D0.

The cup product onH.EIQ/will be completely determined by the products xixj for i¤j; x_i²; xi; ²:

By (1)–(8) above, these products must have the form xixj DX

k

D_ij^kx_kCEij; x_i²DCX

j

Fijxj CGi;

xiDX

j

IijxjCJi; ²DX

j

Kjxj C!

for some classesD_ij^k; Fij 2H².BIQ/,Eij; Gi; Iij 2H⁴.BIQ/,Ji; Ki2H⁶.BIQ/

and!2H⁸.BIQ/. We can assume also thatD_ij^k is symmetric ini andj. Note that

the classesD_ij^k; : : : ; Ki are uniquely determined becausef1; x1; : : : ; xn; gis a basis forH.EIQ/as anH.BIQ/–module.

Proposition 2.4 We have the identities

Fij DDⁱ_ij for i¤j;

Fi i D0;

Iij DEij for i¤j;

Ii i DGi; KiDJi: Proof We have

ˆ

E=B

x_i²xj D ˆ

E=B

xi

X

k

D_ij^kx_kCEij

DD_ijⁱ : On the other hand,

ˆ

E=B

x_i²xj D ˆ

E=B

xj

CX

k

Fi kxkCGi

DFij: Equating these givesFij DD_ijⁱ fori¤j. Similarly, from´

E=Bx_i³D0, we getFi iD0.

Evaluating ´

E=Bxixj two different ways gives Iij D Eij for i ¤ j, evaluating

´

E=Bx_i² in two different ways gives Ii i DGi, and evaluating ´

E=Bxi² in two different ways givesKi DJi.

After making the simplifications given byProposition 2.4, we have xixj DX

k

D_ij^kx_kCEij; (2-1)

x²_i DCX

j j¤i

Dⁱ_ijxjCGi; (2-2)

xiDGixiCX

j j¤i

EijxjCJi; (2-3)

²DX

j

JjxjC!:

(2-4)

3 Families Bauer–Furuta theory

LetX be a compact, oriented, smooth4–manifold withb1.X /D0. Letsbe a spin^c– structure onX with characteristiccDc1.s/2H².XIZ/. Letd D¹₈.c² .X //be the index of the associated spin^c Dirac operator.

LetS¹act onC by scalar multiplication and trivially onR. As shown by Bauer and Furuta[3], one can take a finite-dimensional approximation of the Seiberg–Witten equations for.X;s/to obtain anS¹–equivariant map

fW.C^a˚R^b/^C!.C^a0˚R^b0/^C

for somea; b; a⁰; b⁰0, wherea a⁰Dd andb⁰ bDb_C.X /. HereT^C denotes the one-point compactification ofT. By construction,f sends the point at infinity in .C^a˚R^b/^Cto the point at infinity in.C^a0˚R^b0/^C. Additionally,f can be chosen so that its restrictionfj.R^b/^CW.R^b/^C!.R^b0/^Cis the map induced by an inclusion of vector spacesR^bR^b0. For the purposes of this paper, it is more convenient to look at the Seiberg–Witten equations onX with the opposite orientation. Once again we obtain anS¹–equivariant map of the form

(3-1) f W.C^a˚R^b/^C!.C^a0˚R^b0/^C; but nowa,a⁰,b andb⁰satisfya⁰ aDd andb⁰ bDb .X /.

The process of taking a finite-dimensional approximation of the Seiberg–Witten equa- tions can be carried out in families[20;2]. LetB be a compact, smooth manifold.

Consider a smooth familyWE !B with fibres diffeomorphic toX and suppose that there is a spin^c–structures_E=B on T .E=B/ which restricts tos on the fibres ofE. Taking a finite-dimensional approximation of the Seiberg–Witten equations for the family E (with the opposite orientation onX), we obtain a family of maps of the form(3-1). More precisely, we obtain complex vector bundlesV andV⁰overB of ranks a anda⁰, real vector bundlesU and U⁰ over B of ranksb andb⁰, and an S¹–equivariant map of sphere bundles

f WSV;U !SV⁰;U⁰

covering the identity on B. Here SV;U and SV⁰;U⁰ denote the fibrewise one-point compactifications ofV ˚U andV⁰˚U⁰. The groupS¹acts onV andV⁰by scalar multiplication and trivially onU andU⁰. The action ofS¹on the direct sumsV˚U andV⁰˚U⁰extends continuously to the fibrewise one-point compactificationsSV;U

andSV⁰;U⁰. Moreover, we have, inK⁰.B/andKO⁰.B/, respectively, V⁰ V DD; U⁰ U DH .X /;

whereD2K⁰.B/is the families index of the family of spin^c Dirac operators onE determined bys_E=BandH .X /is the vector bundle onBwhose fibre overb2Bis the space of harmonic antiselfdual2–forms on the fibre ofE overb(with respect to some

smoothly varying fibrewise metric onE). By stabilising the mapf, we can assume that V andU are trivial vector bundles. As shown in[2], the mapf may be constructed so as to satisfy two further properties. First, we may assume thatU⁰ŠU ˚H .X / and that the restrictionfj^SUWSU !SU⁰ is the map induced by the inclusionU !U⁰. Second, we may assume thatf sends the point at infinity in each fibre ofSV;U to the point at infinity of the corresponding fibre of SV⁰;U⁰. LetBV;U SV;U denote the section at infinity and similarly defineB_V0;U⁰S_V0;U⁰. Thenf sendsB_V;U toB_V0;U⁰. Hence,f defines anS¹–equivariant map of pairs

f W.SV;U; BV;U/!.SV⁰;U⁰; BV⁰;U⁰/:

Theorem 3.1 Suppose that W E !B is a smooth family of simply connected, positive-definite4–manifolds over a compact baseB and that T .E=B/admits a spin^c– structures_E=B. LetD2K⁰.B/denote the index of the family of spin^c–Dirac operators associated tos_E=B. Thencj.D/D0forj > d,whered is the virtual rank ofD.

Proof This result is a variant of[1, Theorem 1.1]. We give a streamlined proof. As explained above, taking a finite-dimensional approximation of the Seiberg–Witten equa- tions for the familyE(with opposite orientation onX), we obtain anS¹–equivariant monopole map fW SV;U !SV⁰;U⁰. Since X is positive-definite,H .X / D0 and U⁰DU. Sof takes the form

fWSV;U !SV⁰;U;

with the property thatfj^SU is the identitySU!SU. We also have thatV⁰ V DD. Let V;U andV⁰;U denote theS¹–equivariant Thom classes ofSV;U andSV⁰;U. Consider the commutative diagram

.SV;U; BV;U/ ^{f //}.SV⁰;U; BV;U/

.SU; BU/

j

OO

id //.SU; BU/

j⁰

OO

By the Thom isomorphism in equivariant cohomology, we must have f.V⁰;U/DˇV;U

for someˇ2H_S^2d₁.BIZ/. On the other hand,j._V;U/De_S1.V /_U and.j⁰/._V0;U/D e_S1.V⁰/U, wheree_S1.V /ande_S1.V⁰/denote theS¹–equivariant Euler classes ofV andV⁰andU is theS¹–equivariant Thom class ofU. Therefore,

ˇe_S1.V /U Dj.ˇV;U/Djf.V⁰;U/D.j⁰/.V⁰;U/De_S1.V⁰/U:

Hence,

(3-2) e_S1.V⁰/Dˇe_S1.V /

for someˇ2H_S^2d1.BIZ/. Note that, sinceS¹acts trivially onB, we haveH_S1.BIZ/Š H.BIZ/Œx, whereH_S1.ptIZ/DZŒx. Using a splitting principle argument, it is easy to see that, ifS¹acts on a complex rankmvector bundleW by scalar multiplication, then

e_S1.W /Dx^mCx^{m 1}c1.W /C Ccm.W /:

Now, by stabilisation, we may assume thatV is a trivial bundle,V ŠC^a. ThenV⁰has the same Chern classes asD. So

e_S¹.V⁰/Dx^a0Cx^{a0 1}c1.D/C Cca⁰.D/; e_S¹.V /Dx^a: Then, writing

ˇDˇ0x^dCˇ1x^{d 1}C Cˇd; equation(3-2)becomes

x^a0Cx^{a0 1}c1.D/C Cca⁰.D/Dˇ0x^dCaCˇ1x^{dCa 1}C Cˇ_dx^a: Then, sincedCaDa⁰, it follows thatˇj Dcj.D/for0j d and thatcj.D/D0 forj > d.

Lemma 3.2 LetP !B be a principal PU.m/–bundle and WE!B the associated CP^{m 1}–bundle. Then the pullback.P /ofP to the total space ofEadmits a lift of structure group toU.m/.

Proof LetG PU.m/be the subgroup of PU.m/fixing a point inCP^{m 1}. Then clearly.P /admits a reduction of structure toG. On the other hand it is easy to see thatG'U.m 1/and that the inclusionG!PU.m/factors through the projection U.m/!PU.m/. Therefore,.P /admits a lift of the structure group toU.m/.

Lemma 3.3 There exists a fibre bundleWF !Bsuch that:

(1) WH.BIQ/!H.FIQ/is injective.

(2) Fori D1; : : : ; n,there exist classesi 2H²..E/IZ/such thati restricted to the fibres of.E/equalsi.

The point of this lemma is that thei areintegralcohomology classes whereas thexi

defined earlier are only rational.

Proof Consider the Leray–Serre spectral sequenceEr^p;q for WE !B. Note that E₂^p;q DE₃^p;q. We have seen thatd3.j/ is zero rationally, but it need not be zero overZ. Therefore,gj Dd3.j/2E₃^3;0DH³.BIZ/for j D1; : : : ; nare all torsion classes. By a result of Serre[12], every torsion class inH³.BIZ/is represented by the lifting obstruction for some principal PU.m/–bundle, where the rankmis allowed to vary. Thus, for i D1; : : : ; n, we can find anmi and a principal PU.mi/–bundle Pi!Bsuch thatgi is the lifting obstruction forPi. LetiWEi!Bbe the associated CP^{mi 1}–bundle. ByLemma 3.2, the pullback ofgi toEi must vanish.

LetFDE1^BE2^{B B}Enand letWF!Bbe the projection. By induction onn, it is straightforward to see thatWH.BIQ/!H.FIQ/is injective. Moreover, .gi/D0for alli. Hence, the Leray–Serre spectral sequence for.E/!Bdegen- erates overZatE2. Hence, fori D1; : : : ; n, there exist classesi 2H²..E/IZ/

such thati restricted to the fibres of.E/equalsi.

Theorem 3.4 Let WE!B be a family with structure groupDiff0.X /. Then ˆ

E=B

e^.1x1CCnxn/=2A.T .E=B//y D0

for all 1; : : : ; n2 f1; 1g.

Proof LetWF !B be as in the statement ofLemma 3.3. SinceWH.BIQ/! H.FIQ/is injective, to show that

ˆ

E=B

e^.1x1CCnxn/=2A.T .E=B//y

is zero, it suffices to show that it pulls back to zero under. Therefore, we may restrict to families with the property that there exists classes1; : : : ; n2H².EIZ/such that i restricted to the fibres ofE equalsi. Let

cD11C12C Cnn2H².EIZ/:

Thenc is a characteristic forT .E=B/in the sense that the mod2reduction ofc is w2.T .E=B//. Therefore, the third integral Stiefel–Whitney class ofT .E=B/vanishes and soT .E=B/admits some spin^c–structures⁰. Letc⁰Dc1.s⁰/2H².EIZ/. Then, sincec⁰is a characteristic forT .E=B/, we must have

c⁰D

n

X

iD1

kiiC./

for some odd integersk1; : : : ; knand some2H².BIZ/. For eachi, letLi !E be the line bundle withc1.Li/Di. The set of spin^c–structures forT .E=B/is a torsor

over the group of line bundles onE. So we may consider the spin^c–structure sDL^a₁¹˝L^a₂²˝ ˝L^a_nⁿ˝s⁰;

whereaiD¹₂.i ki/. It follows thatc1.s/DcC./. Now we applyTheorem 3.1 to the familyE !B equipped with the spin^c–structures. LetD 2K⁰.B/be the families index of this spin^c–structure. Since

c1.s/j^X D.cC.//j^XD11C Cnn;

we find (by the Atiyah–Singer index theorem) that the virtual rank ofDis given by d D¹8..11C Cnn/² n/D¹8.n n/D0:

Therefore,Theorem 3.1says thatcj.D/D02H^2j.BIQ/for allj > 0. So Ch.D/D0.

Now, by the families index theorem, Ch.D/D0D

ˆ

E=B

e^c1.s/=2A.T .E=B//y D0:

To finish, we observe that, sincexijX Di DijX, it follows thati DxiC.i/ for somei 2H².BIQ/. Therefore,

c₁.s/D₁x₁C₂x₂C _nx_nC.C₁₁C C_nn/ and hence

Ch.D/D0D ˆ

E=B

e^{.1x1C2x2CnxnC.C11CCnn//=2}A.T .E=B//y De^.C11CCnn/=2

ˆ

E=B

e^{.1x1C2x2Cnxn/=2}A.T .E=B//:y

Multiplying through bye ^.C11CCnn/=2, we obtain the theorem.

Theorem 3.5 Let W E !B be a family with structure group Diff0.X /. Fix i 2 f1; : : : ; ngand,for eachj ¤i with1j n,let j 2 f1; 1gbe given. Set

cD3xiCX

j j¤i

jxj:

Then ˆ

E=B

e^c=2A.T .E=B//y De^u;

whereu2H².BIQ/is given by uD

ˆ

E=B 1

48c³ ₄₈¹p1c:

Proof As in the proof ofTheorem 3.4, it suffices to prove the result for families with the property that there exist classes1; : : : ; n2H².EIZ/such thati restricted to the fibres ofEequalsi. Let

cD3xiCX

j j¤i

jxj:

Arguing as in the proof of Theorem 3.4, there exists a spin^c–structures such that c1.s/DcC./for some 2 H².BIQ/. We applyTheorem 3.1to the family E!Bequipped with the spin^c–structures. LetD2K⁰.B/be the families index of this spin^c–structure. Since

c1.s/j^X D31CX

j j¤i

jj;

we find that the virtual rank ofDis given by d D¹₈..3iCX

j j¤i

jj/² n/D¹₈.9C.n 1/ n/D⁸₈D1:

Therefore,Theorem 3.1says thatcj.D/D02H^2j.BIQ/for allj > 1. Using the Newton identities and the fact thatDhas virtual rank1, we find that

Ch.D/De^c1.D/: Now, by the families index theorem,

Ch.D/De^c1.D/D ˆ

E=B

e^c1.s/=2A.T .E=B//y D ˆ

E=B

e^.cC.//=2A.T .E=B//:y Therefore,

(3-3)

ˆ

E=B

e^c=2A.T .E=B//y De^u;

whereuDc1.D/ ¹₂./. Equating degree2components in(3-3), we find that uD

ˆ

E=B 1

48c³ ₄₈¹p1c:

4 Cohomology rings of families

In this section we apply Theorems 3.4 3.5 to obtain constrains on the structure of the cohomology ringH.EIQ/ and on the characteristic classesp1 ande. These

constrains will then be used inSection 5to deduce various properties of the tautological classes ofE.

Proposition 4.1 Let WE ! B be a family with structure group Diff0.X /. The following identities hold:

0D ˆ

E=B

xixjxi .for distincti; j; k/;

(4-1) 0D

ˆ

E=B

x³_i C3X

j j¤i

xix_j² p1xi

.for eachi /;

(4-2)

0D ˆ

E=B

xix_j³Cx_i³xj C3 X

k¤ki;j

xixjx²_k p1xixj

.for distincti; j /;

(4-3)

0D ˆ

E=B

X

i

x_i⁴C6X

i;j i <j

x_i²x_j²C3e² 2p1

X

i

x_i² (4-4) :

Proof Theorem 3.4gives ˆ

E=B

e^.1x1CCnxn/=2A.T .E=B//y D0:

Expanding the exponential and integrating, we see that the degree2mcomponent of the left-hand side has the form

X

jIjmC2

I˛m;I

for some cohomology classes˛_m;I 2H^2m.BIQ/, where the sum is over subsets of f1; 2; : : : ; ngof sizemC2and

I DY

i2I

i:

Each_I can be thought of as a function_IW f1; 1gⁿ!Q. Thought of this way, the fIg^I are linearly independent. Indeed they are the characters ofZⁿ₂; namely,I is the character of the1–dimensional representation in which thei^th generator ofZⁿ₂ acts as

1ifi