Greg Friedman

Singular chain intersection homology for traditional and super-perversities

We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coecients, and with traditional perversities or superperversities (those satisfying ¯ p(2) > 0). For the case ¯ p(2) = 1, these latter perversitie were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptability to broader classes of stratified spaces. 2000 Mathematics Subject Classification: Primary: 55N33; Secondary: 32S60, 57N80

Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

Abstract In this paper, we develop Leray–Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E 2 terms of the spectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this program, we establish the properties of stratified fibrations over unfiltered base spaces and of their mapping cylinders. We also prove a folk theorem concerning the stratum-preserving homotopy invariance of intersection homology.

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.

The symmetric signature of a Witt space

Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miscenko for manifolds. Our construction has all of the expected properties, including invariance under stratified homotopy equivalence.

Additivity and non-additivity for perverse signatures

Abstract A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any oriented stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Walls non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface, Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegels version of Novikov additivity for Witt signatures is a special case of this corollary.

Triangulations of 3–dimensional pseudomanifolds with an application to state-sum invariants

We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangula- tions, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also provide an application to state-sum invariants of 3-dimensional topological pseudomanifolds. AMS Classification 57Q15, 57Q25; 57N80, 57M27

Chapter 4 – Knot Spinning

Publisher Summary There are many important ways to construct higher dimensional knots. This chapter focuses on algebraic knots, also examining the links of singularities of complex algebraic varieties. The chapter explains spinning constructions for higher-dimensional knots. These methods for creating new knots from knots of lower dimension can be described geometrically, and so spun knots often can be visualized directly (or at least schematically). The chapter then describes the basics of higher-dimensional knot theory and then turns to specific constructions, including simple spinning, superspinning, twist spinning, frame spinning, rolling, and deform spinning. It then discusses some new hybrid constructions, such as frame twist spinning, frame deform spinning, and frame rolling. Examples of important historical applications and results are illustrated.

Unitary equivalence of normal matrices over topological spaces

Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point of X. We use obstruction theory to establish a necessary and sufficient condition for A and B to be unitarily equivalent. We also determine bounds on the number of possible unitary equivalence classes in terms of cohomological invariants of X.

Superstrings, Geometry, Topology, and *-algebras

This volume contains the proceedings of an NSF-CBMS Conference held at Texas Christian University in Fort Worth, Texas, May 18-22, 2009. The papers, written especially for this volume by well-known mathematicians and mathematical physicists, are an outgrowth of the talks presented at the conference. Topics examined are highly interdisciplinary and include, among many other things, recent results on D-brane charges in

All frame-spun knots are slice

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