Showing papers in "Commentarii Mathematici Helvetici in 1993"

Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms

Abstract: A quadratic differential on a Riemann surface M determines certain " topological" data: the genus o f M; the orders o f zeros and poles; and the orientability o f the horizontal foliation. In this note we determine which collections o f data can be realized by quadratic differentials with finite area. A pseudo-Anosov diffeomorphism of M also determines certain topological data: the genus of the M; the types of the singularities and the orientability of the stable foliation. As a corollary to our result on quadratic differentials we determine which topological data can be realized by pseudo-Anosov diffeomorphisms on oriented surfaces. Let X be a closed Riemann surface o f genus g with a system of holomorphic coordinate charts { U,., h,. }. This means that { U, } is a covering of X by open sets; h,. is a homeomorph i sm of U,. to an open set in the complex plane and hu ~ h; ~ is conformal whenever defined. Let q be a positive integer. A meromorphic q-differential 05 on X is a set o f meromorphic function elements 05,, in the local parameters zv = h,.(p) for which the t ransformat ion law

Classical knot and link concordance

Abstract: In [G1], we combined the slicing obstructions of Levine [L] with those of Casson and Gordon [CGI], [CG2], [Go] in a nontrivial way. Essentially we related the metabolizer for the Seifert form of a slice knot which Levine guaranteed to the characters on the homology of the 2-fold branched cover (with vanishing Casson Gordon invariants) which Casson and Gordon guaranteed. In this paper we will simultaneously consider all the q-fold branch covers for q a prime power. We will define F + which strengthens the group F ' of [GI]. In [GL2] Livingston and Gilmer applied these methods to concordances of two component links with linking number zero. They defined an algebraic group ~ which enhanced F' to study concordances of links with two components. In this paper we will define an analogous strengthening ~ ' of ~, We also calculate enough Casson-Gordon invariants to decide when a genus one knot maps to zero in F +. Finally we give a new example of a link which is a fusion of a boundary link but not concordant to a boundary link. Cochran and Orr [CO] were the first to discover such links. Livingston was the first to observe that our work on Casson Gordon invariants could be applied to this problem [Li]. We would like to thank Chuck Livingston for many valuable conversations. We work in the smooth category.

Relative growth rates of closed geodesics on a surface under varying hyperbolic structures

Abstract: This paper has two interrelated goals. The first is to give necessary and sufficient condi t ions that an ac t ion o f a surface group on a tree be geometric, that is to say, be dual to a cod imens ionI measured lamina t ion on the surface. The second is to s tudy the l imit ing rat ios of lengths o f simple closed geodesics under a degenera t ing sequence o f hyperbol ic structures on the surface. We begin by explaining in more details each o f these goals. In o rder to explain the first, let us return to an old result o f Stallings. Recall f rom [5] that if X c M is a compac t codimension-1 submani fo ld , then there is an act ion of ~ ( M ) on a simplicial tree dual to X c M (or more precisely dual to )~ c 3~t where ~ t is the universal covering of M).

On Cheeger's inequality

Abstract: In Ch], Cheeger proved the following general lower bound for the rst eigenvalue 1 of a closed Riemannian manifold: Theorem ((Ch]): 1 1 4 h 2 ; where h = inf N area(N) min(vol(A); vol(B)) where N runs over (possibly disconnected) hypersurfaces of M which divide M into two pieces A and B, and where area denotes (n ? 1)-dimensional volume, and vol denotes n-dimensional volume, where n = dim(M). h(M) is called the Cheeger constant of M. Cheeger's inequality is quite straightforward to prove, and is essentially the co-area formula of geometric measure theory. It is therefore surprising that the inequality plays such a crucial role in the study of the geometry of the Laplace operator, see Bu3]. Indeed, one has the following general upper bound for 1 in terms of h, due to Peter Buser Bu]: where c 1 ; c 2 depend only on a lower bound on the Ricci curvature of M. Thus, from a qualitative point of view, 1 and h are essentially the same thing, in the sense that one tends to zero if and only if the other does (in the presence of bounded curvature). We observe that Cheeger's inequality is true, and is proved in exactly the same way, when M is a complete, non-compact manifold, or a manifold with boundary and either Dirichlet or Neumann boundary conditions, provided one interprets 1 , and h correctly. It has therefore been an interesting question to understand, in a general way, how sharp Cheeger's inequality really is. A major problem in coming to terms with this question has been that, for the most part, Cheeger's inequality is the only generally useful method known for estimating 1 from below. In this paper, we will explore this question in three ways. First of all, by 1 a celebrated theorem of Selberg Se], there are general lower bounds 1 (S p) 3 16 for certain arithmetic Riemann surfaces S p , which we will discuss below. Selberg raised the question of whether 1 (S p) 1 4 for these surfaces, and it was suggested in Bi] that perhaps one could demonstrate this by showing that h(S p) 1 for these surfaces. We will show that this is not the case, and indeed h(S p) is so small for these surfaces that one cannot even obtain Selberg's 3 16 bound via Cheeger's constant: Theorem 1.1: For p 1(mod …

Gluing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities

Abstract: This paper deals with the problem of characterizing quasihomogeneous isolated singularities The history begins in 1971 with the beautiful result of Saito [22]: an isolated complex hypersurface singularity with defining equation f is quasihomogeneous (ie, after a change of coordinates f can be made into a quasihomogeneous polynomial) if and only iff ~j(f), where j(f) is the ideal generated by the partial derivatives off (this ideal is also called the jacobian ideal off) In the subseqeunt years this result was extended to other fields and significantly generalized in papers by Scheja and Wiebe, see [24], [25] and [26] Among other powerful results they showed that a complete intersection (R, m, k) with isolated singularity is quasihomogeneous if and only if there exists a k-derivation 6 of R which induces an isomorphism on the Zariski tangent space m/m 2 If dim R = 2, then the assumption that R is a complete intersection can be discarded and the requirement on the derivation 6 can be weakened: it suffices that 6 induces a nonnilpotent transformation of m/m 2 A concise account of their work can be found in Platte's paper [21] In 1985, Wahl [29] characterized quasihomogeneous Gorenstein surface singularities in terms of certain invariants associated with the resolution of singularities There the aforementioned criterion of Scheja and Wiebe was used In 1984, Kunz and Waldi [15] characterized quasihomogeneous reduced Gorenstein algebroid curves over an algebraically closed field k of characteristic 0 by the condition that the cokernel R/J of the canonical homomorphism from the (universally finite) module of K~ihler differentials to the module of regular differentials of R/k is Gorenstein If R is a complete intersection then J is the K~ihler different of R/k, ie, the ideal generated by the maximal minors of the jacobian matrix In 1987 the second author noticed in his thesis [16] the relevance of maximal Cohen-Macaulay modules for the problem ofquasihomogeneity He conjectured that

Hyperbolic volume and modp homology

Abstract: IfM is a closed, orientable hyperbolic 3-manifold such that\(\dim _{Z_p } \)H1 (M;Z p ) >-for some primep, thenM contains a hyperbolic ball of radius (log 5)/4. There is also a related result in higher dimensions.

Transcendental submanifolds of Rn

Abstract: In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set is related to the real algebraic cycles of the real part. As a corollary we give examples of smooth submanifolds of a Euclidean space which can not be isotoped to real parts of complex nonsingular subvarieties in the corresponding projective space.

A classification of polynomial algebras as modules over the Steenrod algebra

Abstract: Suppose that A2 is the mod-2 Steenrod algebra, and that R = F2[x1, . . . , xn] is the polynomial ring in n variables over the prime field F2. Campbell and Selick [3] observed that the equations Sqxi = x 2 i−1 for 2≤ i≤ n, and Sqx1 = x 2 n, define an action of A2 on R that makes R isomorphic as an A2-module to the A2-module defined by the standard action on R. In that paper, they also pose the following question, due to Tom Hunter: does the equation Sqxi = ∑ jmijx 2 j (where M = [mij ] is any n-by-n matrix over F2) define an action of A2 on R? In this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V ) be the symmetric

Homotopy and isotopy in dimension three

Abstract: Let M be a closed p2-irreducible 3-manifold. It is a long standing problem to decide if homotopic homeomorphisms of M must be isotopic. The answer is now known to be affirmative if M is Haken, [Wal], see also [L], or if M is a Seifert fiber space [Ho-R] [Bon] [B-R] [A] [R] [Scl] [B-O], and for a few other special manifolds [B-R]. Thus is now seems reasonable to conjecture that the answer is always affirmative. However, if one considers reducible manifolds, there is a counter example [F W]. In this paper, we further enlarge the class of 3-manifolds for which the above conjecture can be proved. If a closed P2-irreducible 3-manifold is non-orientable, it must be Haken, so we consider only orientable 3-manifolds in the rest of this paper. Let M be an orientable 3-manifold, let F be a closed orientable surface not S 2 and let f: F ~ M be an immersion which injects n~ (F). Let Mr denote the cover of M such that nj (Mr) equals f,(nl(F)) and let M denote the universal cover of M. We will suppose that the lift of f into MF is an embedding. (Note that this is automatic iff is least area in the smooth or PL sense.) Thus the pre-image in M of f(F) consists of an embedded plane /7 which covers F in M,, and the translates of // by hi(M). We will say that f has the k-plane property if, given k distinct translates of H, some pair is disjoint. In this paper we will consider the case when k equals 3. A map with the 3-plane property has no transverse triple points. We will say that f has the 1-line-intersection property if two distinct translates of /7 are disjoint or intersect transversely in a single line. The main result of this paper is THEOREM 1.1. Let M be a closed orientable irreducible 3-manifold which is neither Haken nor a Seifert fiber space. If there is a closed orientable surface F, not S z, and an immersion f:F~M which injects nl(F) and has the 3-plane and l-line-intersection properties, then homotopic homeomorphisms of M are isotopic. In [H-S], we show that if M satisfies the hypotheses of this theorem, and M is homotopy equivalent to an irreducible 3-manifold N, then M and N are homeomor