Intersection Pairing Research Articles

A vanishing theorem for Seiberg-Witten invariants

It is shown that the quotients of Kahler surfaces under free anti-holomorphic involutions have vanishing Seiberg-Witten invariants. Various vanishing theorems have played important roles in gauge theory. The first among them, due to S. K. Donaldson [2], states that the Donaldson invariants vanish for any smooth closed oriented 4-manifold X which decomposes to a connected sum X1#X2 with b2 (X1) > 0, b + 2 (X2) > 0. (Here b + 2 (Xi) is the dimension of a maximal subspace of H2(Xi,Z) on which the intersection pairing is positive definite.) E. Witten [12] has shown more recently that such a connected-sum manifold has also vanishing Seiberg-Witten invariants. Even though its proof is simple, Witten’s vanishing theorem is equally useful as Donaldson’s vanishing theorem; for example, it implies that any symplectic 4-manifold cannot be decomposed into the above connected sum by combining with Taubes’ theorem in [9]. In this note we show that Seiberg-Witten invariants vanish for another class of 4manifolds. These manifolds are obtained in connection with real algebraic geometry, and the construction was originally proposed in Donaldson [3]. See Remark 4 below. To state our theorem, recall that a map σ between two almost complex manifolds is called anti-holomorphic if σ∗J1 = −J2σ∗ on the tangent bundles, where Ji are the almost complex structures of the manifolds. In the following, K denotes the canonical bundle of an almost complex manifold (underlying a Kahler manifold). Theorem 1. Let X be a Kahler surface with Ke X > 0 and b2 (X) > 3. Suppose that σ : X → X is an anti-holomorphic involution without fixed points. Then the quotient manifold X = X/σ has vanishing Seiberg-Witten invariants. In view of Witten’s vanishing theorem, it is natural to examine whether the quotient X can be decomposed into a connected sum: Proposition 2. If X is a simply-connected Kahler surface or more generally symplectic 4-manifold with b2 (X) > 1 and σ : X → X is a free involution, then the quotient manifold X = X/σ is not diffeomorphic to any connected sum X1#X2 with both b2 (Xi) > 0. 1991 Mathematics Subject Classification. Primary 57R55, 57R57, 57N13; Secondary 14P25. Work supported by the Research Board grant of University of Missouri. Typeset by AMS-TEX 1

Simply-connected 4-manifolds with a given boundary

Let M be a closed, oriented, connected 3-manifold and let ( Z n , L) be a symmetric bilinear form which presents H ∗( M; Z ). Lt V L ( M) be the set of all oriented homeomorphism types of compact, 1-connected, oriented 4-manifolds with boundary M and intersection pairing isomorphic to ( Z n , L). We will give a complete description of the sets V L ( M).

The Torelli groups for genus 2 and 3 surfaces

THE purpose of this paper is to show that the genus 2 Torelli group is free on infinitely many Dehn twists on separating curves. Moreover, the set of free generators can be identified with the set of splittings of the homology of a genus 2 surface into two subspaces mutually orthogonal and unimodular with respect to the intersection pairing. In addition, it is shown that the third integer homology of the genus 3 Torelli group naturally contains an infinitely generated free abelian group. This is a permutation module for the symplectic modular group. The method used is a study of the period mapping to Siegel space. In Sections 2 and 3 we review some background material. Section 4 shows that the genus 2 Torelli group is free. Section 5 contains additional background material and the result on the third homology. Section 6 gives a homological application.

Two dimensional gauge theories revisited

Two dimensional quantum Yang-Mills theory is reexamined using a non-abelian version of the Duistermaat-Heckman integration formula to carry out the functional integral. This makes it possible to explain properties of the theory that are inaccessible to standard methods and to obtain general expressions for intersection pairings on moduli spaces of flat connections on a two dimensional surface. The latter expressions agree, for gauge group SU (2), with formulas obtained recently by several methods.

Periodic maps on simply connected four-manifolds

IN THIS paper we describe certain invariants of periodic maps on closed, simply connected four-manifolds with the purpose of classifying such maps up to homeomorphism. The manifolds themselves are classified by Freedman [9] in terms of the intersection pairing on second homology and the Kirby--Sicbcnmann invariant. Gcncralizing Freedman’s result to some nonsimply connected four-manifolds Hambleton and Kreck [IO] have recently shown that in the case of a cyclic fundamental group of odd order the manifold is determined up to homeomorphism by the same two invariants togcthcr with the fundamental group (with the intcrscction pairing being understood now as dcfincd on the torsion free part of the second homology group). A similar statement for cvcn order groups fails to bc true. although its failure is less dramatic for algebraic surfaces as follows from the work by the same authors [I I]. From the point of view of the fundamental group acting on the universal cover these results concern the topological classification of fret periodic maps on simply conncctcd four-manifolds. In gcncral, howcvcr. a periodic map is allowed to have fixed points or periodic points of period smaller than the map period. When this happens, the quotient orbifold has singularities. We shall show that in the cast of isolated singularities one can still classify periodic maps by means of some invariants associated with this orbifold, although in the singular case one must also take into account the torsion part of its second homology. In the more rigid situation of a free periodic map no torsion information is needed. This can bc partially accounted for by the fact that many four-manifolds do not support any such maps. On the other hand, by a result of Edmonds [5]. every closed, simply connected fourmanifold supports infinitely many periodic maps with isolated fixed points. The paper is organized in four sections. In Section 2 we discuss the homotopy classification of locally linear, pseudofree actions of finite groups on simply connected fourmanifolds. There we introduce some terminology and prepare the obstruction theory needed for the results of the next two sections. Our main result is the topological classification of semifree periodic maps of odd period. This is the content of Theorem 3.1. A related but somewhat weaker result for periodic maps of even period is contained in Theorem 3.2. As an application of our techniques, we prove in Section 4 that any locally linear, pscudofree action of a finite group on the complex projective plane is topologically conjugate to a linear action. This last statcmcnt about actions on CP’ has also been asserted in [I93 in the context of finite group actions on the Chern manifold. There we concentrated our efforts on actions without fixed points referring the reader for the proof in the semifrce case to the combined results of [7] and [ 121. This point requires now further elaboration for maps of composite period as the authors of [ 121 have meanwhile added to their result a technical hypothesis of

Involutions on odd four-manifolds

It is shown that every closed, simply connected topological 4-manifold having an odd intersection pairing, with the possible exception of the fake C P 2, admits an involution. We show that in many cases the involutions described here can be constructed to be locally linear, provided the Kirby-Siebenmann triangulation obstruction vanishes. It remains an open question, reduced to a question about characteristic elements of quadratic forms, whether this is true in general.

On the statistical properties of estimated rotations

This paper discusses theorems from the mathematical statistics literature on confidence regions for an estimated rotation. The theorems can be used to analyze pairs of homologous intersections of fracture zones with magnetic anomaly lineations on opposing plates. Two classes of probability models on the data are considered: The “fixed u” and the “random u” models, corresponding to, in the geophysical context, whether error is attributed to the measurements on one or on both plates. For each model, large sample and concentrated error approximate confidence regions are given. It is found that the geometric features of the confidence regions are in rough agreement with the analysis of partial uncertainty rotations by Stock and Molnar. The paper concludes with some rudimentary small sample, diffuse error distribution theory. For illustrative purposes, a data set from the Gulf of Aden is repeatedly analyzed using a variety of probabilistic models.

Equivariant intersection forms, knots in 𝑆⁴, and rotations in 2-spheres

We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in S 4 {S^4} with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.

Quadratic functions and smoothing surface singularities

LET (X, 0) be an isolated complex analytic surface singularity which is reduced and irreducible. This paper is concerned with certain necessary conditions (X, 0) must satisfy in order that it admits a smoothingf: ( ?Z-, 0) + (A, 0); so,fis a flat morphism of analytic germs with smooth general fibre and X as special fibre. The general fibre of such an f may be represented by a Stein manifold of complex dimension 2, which is the interior of a compact 4manifold with boundary (M, dM); M is called the Milnorjbre of the smoothing. Moreover, there is a diffeomorphism of ZM onto a link L of (X, 0) (i.e. the intersection of X with a small sphere about 0 in an ambient space), which is well-defined up to isotopy. So, L may be regarded as the boundary of both M and a resolution .? of X. An important invariant of M is its middle homology group Hz(M) (which is free) equipped with the intersection product (which is symmetric). Note H2 (2) is also free and has a natural set of generators (the fundamental classes of the irreducible exceptional curves); the intersection pairing is negative-definite, but not necessarily even (even if X is a hypersurface). One would like to say as much as possible about the inner product lattice H2 (M) in terms of X or 2. For instance, if X is a Gorenstein singularity, then rk HI (M) = 0 [ll], the pairing on H2 (M) is even [38], and the Sylvester invariants (p,, , p + , p _ ) of the corresponding real pairing can be expressed in terms of x ( [41]: here, ~1 = p, + p+ + p- = rk H, (M)). Our principal contribution is the construction of natural and compatible “quadratic functions” on Hz (M), Hz(x), and H,(L), (where r means torsion subgroup). The construction and its properties are of a topological nature; we therefore need some topological discussion. If L is any compact 3-manifold, then there exists a natural symmetric non-degenerate linking pairing I: H, (L), x H, (L), + Q/Z defined geometrically [39]. Motivated by a result of Morgan-Sullivan [27], we prove that there is a finer object than 1. The singularity determines a homotopy class 9 of complex structures on 5‘ @ iw, (the tangent bundle of L plus a trivial bundle) whose first Chern class is torsion. Then (first part of Theorem 3.7) 9 determines a natural function 4 : H 1 (L), +Q/H such that 4(5*+9)-4(5)-4(rl)=1(5,~). Since 1 is bilinear, we call 4 a quadratic function; if furthermore 4 (no = n2 4 (<) (n E Z), then 4 is called a quadratic form. The function 4 derives its main interest from a universal property with respect to the intersection pairings of compact almost-complex 4-manifolds which have

Simply-connected 4-manifolds with a given boundary

Let M M be a closed, oriented, connected 3 3 -manifold. For each bilinear, symmetric pairing ( Z n , L ) ({{\mathbf {Z}}^n},\,L) , our goal is to calculate the set V L ( M ) {\mathcal {V}_L}(M) of all oriented homeomorphism types of compact, 1 1 -connected, oriented 4 4 -manifolds with boundary M M and intersection pairing isomorphic to ( Z n , L ) ({{\mathbf {Z}}^n},\,L) . For each pair ( Z n , L ) ({{\mathbf {Z}}^n},\,L) which presents H ∗ ( M ) {H_ \ast }(M) , we construct a double coset space B L t ( M ) B_L^t(M) and a function c L t : V L ( M ) → B L t ( M ) c_L^t:{\mathcal {V}_L}(M) \to B_L^t(M) . The set B L t ( M ) B_L^t(M) is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of H 1 ( M ) {H_1}(M) by two naturally occuring subgroups. When ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is an even pairing, we construct another double coset space B ^ L ( M ) {\hat B_L}(M) , a function c ^ L : V L ( M ) → B ^ L ( M ) {\hat c_L}:{\mathcal {V}_L}(M) \to {\hat B_L}(M) and a projection p 2 : B ^ L ( M ) → B L t ( M ) {p_2}:{\hat B_L}(M) \to B_L^t(M) such that p 2 ⋅ c ^ L = c L t {p_2} \cdot {\hat c_L} = c_L^t . Our main result states that when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even the function c ^ L {\hat c_L} is injective, as is the function c L t × Δ : V L ( M ) → B L t ( M ) × Z / 2 c_L^t \times \Delta :{\mathcal {V}_L}(M) \to B_L^t(M) \times {\mathbf {Z}}/2 when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is odd. Here Δ \Delta is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets V L ( M ) {\mathcal {V}_L}(M) are finite and of an order bounded above by a constant depending only on H 1 ( M ) {H_1}(M) . We also show that when H 1 ( M ; Q ) ≅ 0 {H_1}(M;{\mathbf {Q}}) \cong 0 and ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is even, c L t = p 2 ⋅ c ^ L c_L^t = {p_2} \cdot {\hat c_L} is injective. It seems likely that via the functions c L t × Δ c_L^t \times \Delta and c ^ L {\hat c_L} , the sets B L t ( M ) × Z / 2 B_L^t(M) \times {\mathbf {Z}}/2 and B ^ L ( M ) {\hat B_L}(M) calculate V L ( M ) {\mathcal {V}_L}(M) when ( Z n , L ) ({{\mathbf {Z}}^n},\,L) is respectively odd and even. We verify this in several cases, most notably when H 1 ( M ) {H_1}(M) is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1 1 -connected 4 4 -manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an m &gt; 0 m &gt; 0 , depending only on H 1 ( M ) {H_1}(M) , such that for any self-homeomorphism f f of M M , f m {f^m} extends to a self-homeomorphism of any 1 1 -connected, compact 4 4 -manifold with boundary M M .

Pairs of complete intersections

With a local noetherian ring A and with an ideal Z the question may be asked whether the rings A and A/Z are simultaneously complete intersections (that means: complete intersections in the strict sense, after completion if necessary). Special cases are known and the answer is given by means of the corresponding Koszul homology, more precisely by means of the A/Z-modules Hi and Hz. When the ring A/I is a field, according to E. Assmus the ring A is a complete intersection if and only if ZZ2 is equal to the square of ZZ, and when the ring A is regular, according to T. Gulliksen the ring A/Z is a complete intersection if and only if Hr is free (see [2, Theorem 2.71 and [3, Theorem 6.11). Let us consider a system of generators (Y = ((xt, . . . , a,) of the ideal Z (the system is not necessarily minimal). Then let us consider

Heights and Arakelov's Intersection Theory

Introduction. This paper relates the Neron-Tate height on Jacobian varieties to an intersection theory on arithmetic surfaces constructed by S. Arakelov [A1 ]. Here I extend and prove a statement of Arakelov to the effect that his intersection pairing is a Neron pairing [A2I. I use this to prove an analogue of the Hodge Index theorem for arithmetic surfaces, a result due independently to G. Faltings. The results of this paper have been extracted from my thesis, written under the direction of Michael Artin. It differs from the thesis in three ways: Firstly it employs the language of local height functions as defined by A. Neron and J. Tate and expounded by S. Lang [LI ]. In the thesis I used Neron's language of local intersection pairings [N1 ]. The second major difference is in the proof of Theorem 1.3. In my thesis I proved this result by using work of Raynaud and Picard Functors [R 1 ], this method was suggested by M. Artin. In this paper I provide a simpler proof observed by S. Lang (and independently) by B. Gross. Finally the development of Arakalov's theory in the context of Neron Functions in section 2 is due to S. Lang, who has kindly given me leave to publish it. This exposition of my results is due in considerable part to S. Lang; it is essentially an edited transcript of a series of conversations between us, which he provided. It is my great pleasure to thank both him and Michael Artin for their help and encouragement. 1. The Neron pairing. Neron [Ne] has given a pairing between divisors and points on abelian varieties, and also on arbitrary varieties, under suitable conditions. We shall give a complement to the Neron theory on curves. Let C be a curve, by which we mean a complete regular curve, geometrically irreducible over a field K with an absolute value v. Let

2-Sphere Bundles Over Compact Surfaces

Closed 4-manifolds which fiber over a compact surface with fiber a sphere are classified, and the fibration is shown to be unique (up to diffeomorphism). It is well known that there are at most two orientable 4-manifolds which fiber over a given compact surface with fiber the 2-sphere S2. (There is exactly one if the surface has nonempty boundary, and two if it is closed.) If the orientability condition is dropped, then the situation becomes more involved. In particular the (mod 2) intersection pairing is no longer sufficient to distinguish among the mani- folds that arise. One must also consider the ?Tl-action on g2 and the peripheral structure. The purpose of this note is to classify all 4-manifolds (orientable or not) which are total spaces of S2-bundles over compact surfaces. We shall work in the smooth category. Since Diff(S2) deformation retracts to 0(3), we may assume that all bundles that arise have 0(3) as structure group. Along the way it is shown that the bundle structures are unique. That is, if any two 4-manifolds, fibered as above, are diffeomorphic, then there is a fiber preserving diffeomorphism between them which is orthogonal on fibers. Our interest in S2-bundles arose in the study of Lie group actions (in particular of S0(3)) on 4-manifolds. The results obtained here are used in the equivariant classification of such actions (MP).

Spatial knowledge of young and elderly adults: scene recognition from familiar and novel perspectives.

Thirty young adults (mean age = 25.3) and 30 elderly adults (mean age = 65.3) were tested on a memory task in which they were asked to recognize environmental scenes from familiar and novel perspectives. Participants initially viewed slides of 10 business and 10 residential street intersections. Pairs of intersections were then presented and subjects were asked to select the intersection viewed previously. During the recognition phase subjects saw the intersections from the original perspective (0 degrees), rotated 90 degrees from the original perspective, or rotated 180 degrees from the original perspective. Young adults were more accurate than elderly adults and accuracy was greater for business than residential scenes at both age levels. Subjects were more accurate in the 0 than 180 degree condition, while performance in the 90 degree condition was significantly less accurate than in the other two conditions. These results indicate that (1) young adults have better recognition memory than elderly adults for real world scenes, and (2) environmental differentiation aids recognition memory for spatial locations.

Some Routes to Chaos from Limit Cycle in the Forced Lorenz System

Previously we have investigated the response of the Lorenz system under periodic perturba­ tion. In this paper, we focus on the transition from a limit cycle to chaos in the same system. The following three cases of the transition are studied: (I) Chaos through cascades of subharmonic bifurcation. In the periodic region, the stroboscopic mapping in the system is approximated by the Henon mapping. We obtain the Feigenbaum constant a = 4. 715. (II) Chaos due to the production of a homoclinic intersection in the stroboscopic picture. The system shows Intermittency. (III) Chaos due to the production of a pair of heteroclinic intersections in the stroboscopic picture. The system shows Intermittency. The resultant strange attract or has a positive two­ dimensional Lyapunov characteristic number. Power spectra and Lyapunov characteristic numbers are obtained numerically in each case. Furthermore, in connection with Case (II), we show that in a system which has a certain type of spatial symmetry such as is found in the Lorenz model, a symmetric solution cannot undergo subharmonic bifurcation.

The signature of smoothings of complex surface singularities

Let f : (1~3, 0)-"~(t~, 0) be the germ of a complex analytic function with an isolated critical point at the origin. For e > 0 suitably small and 6 yet smaller, the space V ' = f l ( 6 ) ~ D , (where D~ denotes the closed disk of radius e about 0) is a real oriented four-manifold with boundary whose diffeomorphism type depends only on f It has been proved that V' has the homotopy type of a wedge of two-spheres; the number p o f two-spheres is readily computable. Recently an interesting formula for g was given in terms of analytic invariants of a resolution of the singularity at 0 of the complex surface f-1(0) [13]. This formula is proved by applying the Riemann-Roch theorem to the projective completions o f f l ( 0 ) and fi (6) , then canceling terms coming from the parts away from the origin. The purpose of this paper is to find a similar formula for the signature of the intersection pairing on the two-dimensional homology of the manifold V', using the Hirzebruch signature theorem instead of the Riemann-Roch theorem. Various other signature formulas are known, in higher dimensions as well as in dimension two. For f (x , y, z) of the form g(x, y) + z 2, the intersection pairing of V' is the same as a symmetrized Seifert matrix of the compound torus link {g(x, y )=0}~S 3. There is a simple formula for the signature of the symmetrized Seifert matrix of a compound link of one component [20]; hence if g-1(0) is irreducible, there is a simple formula for the signature of V' in terms of the Puiseux pairs ofg. If g1(0) has several branches at the origin, it is possible to find a Seifert matrix for the link defined by g and compute the signature [17], but this process is tedious for all but the simplest links. Formulas also exist for the signature when f(x, y, z) is of the type x a + yb + z c [10], or when f is weighted homogeneous [22]. There is in addition a formula for the signature in terms of mixed Hodge structure [233. The genus (or geometric genus) of the singularity f 1(0) (assumed Stein) is the dimension of HI(V,, C~), where ~" is a resolution of f-1(0). By combining the formulas for the signature and the number # it is possible to show that twice the genus of the singularity f 1(0) is equal to the number of positive plus the number of

A Characterization of Homology Manifolds

A homology «-manifold is a space which has the same local homology at each point as Euclidean n-space. The principal result of this paper is the characterization of triangulable homology manifolds by a global property: The rc-circuit X is a homology manifold if and only if the diagonal cycle A in X x X is Poincare dual to a cocycle with support A (Theorem 1). If X is a smooth manifold, this cocycle represents the Thorn class of the tangent bundle of X. The homological properties of Thorn classes have been studied by Milnor [13; §11] and Spanier [14; Chapter 6]. The proof is based on their techniques. A corollary of this proof is that an w-circuit X satisfies Poincare duality if and only if there is a class U dual to the diagonal which has a certain symmetry with respect to the canonical involution T on XxX; namely U-^V = U^T*V for all V (Proposition 1). Furthermore, for any ^-circuit X, the diagonal cycle is dual to some cocycle U, if coefficients are in a field (Proposition 2). Thus U\(XxX—A) is the obstruction to X being a homology manifold. Propositions 1 and 2 have been obtained independently by P. Holm [8]. The ideas of Lefschetz about intersection theory and the topology of algebraic varieties have been my constant guide (cf. [15]). Theorem 1 can be interpreted in terms of the intersection pairing (Theorem 3). This paper is a revised version of part of my doctoral thesis at Brandeis University [11], written under the supervision of Professor Jerome Levine. I have also been helped by the questions and suggestions of P. Lynch, D. Stone, A. Landman, and especially D. Sullivan. My viewpoint has recently been influenced by the work of I. Fary [3]. Homology will be singular homology throughout, with integer coefficients in §§1 and 2, and field coefficients in §§3 and 4. Sign conventions for products are those of [14].

The intersection pairing on a homogeneous Kähler manifold.

The intersection pairing on a homogeneous Kähler manifold.

The fixed-point property of $(2m-1)$-connected $4m$-manifolds

Suppose that M is a $(2m - 1)$-connected smooth and compact manifold of dimension 4m. Assume that its intersection pairing is positive definite, and denote its signature by $\sigma$. Two notions are introduced. The first is that of a $(\xi ,\lambda )$-map $f:M \to M$ where $\xi \in K(M)$ and $\lambda$ an integer. It describes the concept of f preserving $\xi$ up to multiplication by $\lambda$ outside a point. The second notion is that of $\xi$ being sufficiently asymmetric. It describes in terms of the Chern class of $\xi$ the concept that the restrictions of $\xi$ to the 2m-spheres realizing a basis for ${H_{2m}}(M;Z)$ are sufficiently different so that no map which preserves $\xi$ can move the spheres among themselves. One proves that $(\xi ,\lambda )$-maps with $\xi$ being sufficiently asymmetric have fixed points, except possibly when $\sigma = 2$. On taking $\xi$ to be the complexification of the tangent bundle of M, one sees that mainfolds with sufficiently asymmetric tangent structures have the fixed point property with respect to a family of maps which includes diffeomorphisms. The question of the existence of $(\xi ,\lambda )$-maps as well as the question of the preservation of the fixed-point property under products are also discussed.

Some Aspects of a Switching Policy for Commuter Flow

A wide range of signal switching policies may be adopted for control of traffic flow between a pair of intersections. One of the most important traffic situations is that in which rush-hour commuter traffic travels towards or away from a city center along radial roads. Under these circumstances opposing and tangential flow may be light. Earlier papers have shown that a modified form of the Dunne-Potts switching policy might be applied to pairs of intersections carrying commuter flow. Efficient and flexible operation of the system under a wide variety of flow conditions, and with the minimum of computer surveillance, was demonstrated. The present paper reports the results of further simulation experiments and discussed tentative solutions of some of the practical problems associated with the proposed scheme.