Introduction to Whitney towers

Abstract

These introductory notes on Whitney towers in 4-manifolds, as developed in collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X conference February 2020 in Pisa, Italy. Topics presented include local manipulations of surfaces in 4-space, fundamental definitions related to Whitney towers and their associated trees, geometric Jacobi identities, the classification of order n twisted Whitney towers in the 4-ball and higher-order Arf invariants, and low-order Whitney towers on 2-spheres in 4-manifolds and related invariants. IV-1 Rob SchneidermanSection 2 describes the classification of order n twisted Whitney towers on properly immersed disks in the 4-ball, which illustrates the order-raising intersection-obstruction theory and leads to the formulation of open problems related to certain higher-order Arf invariants which are invariants of classical link concordance. Here the trees associated to Whitney towers are seen to represent invariants in abelian groups related to Milnor's classical link invariants.Section 3 reviews the classical intersection and self-intersection (order 0) homotopy invariants of 2-spheres in a 4-manifold X before introducing order 1 generalizations of these invariants in the setting of Whitney towers. Here new subtleties coming from π 1 X and π 2 X enter the picture. The end of this section describes open problems on the realization of the order 1 invariants when X is closed and π 1 X is non-trivial.The appendix section 4 provides additional material related to Section 2 and Section 3 in the form of outlines and/or details of proofs of results from those sections.Exercises appear at the end of each section.Sections 2 and 3 are largely independent of each other, but both depend on Section 1. Conventions: Manifolds and submanifolds are assumed to be smooth, with generic intersections, unless otherwise specified, and during cut-and-paste constructions corners will be assumed to be rounded. The discussion throughout will also hold in the flat topological category via the notions of 4-dimensional topological tranversality from [13, chap.9]. Orientations will usually be assumed but suppressed unless needed.Acknowledgments: The author is supported by a Simons Foundation Collaboration Grant for Mathematicians. Also thanks to the organizers of the Winter Braids X conference, and of course collaborators Jim Conant and Peter Teichner. 1.11. Twisted trees for twisted Whitney disks 12 1.12. 'Framed tree' terminology 13 1.13. Intersection forests of Whitney towers 13 1.14. Examples 13 1.15. Higher-order Whitney disks and intersections 15 1.16. Order n framed Whitney towers 15 1.17. Order n twisted Whitney towers 16 1.18. Other gradings of Whitney towers 16 1.19. Tree Orientations 17 1.20. 'Moving' unpaired intersections in their trees 18 1.21. Geometric Jacobi Identity in four dimensions 18 1.22. Section 1 Exercises 21 2. Order n twisted Whitney towers in the 4-ball 23 2.1. The order n twisted tree groups 24 2.2. Geometric meaning of th...