Brian A. Munson

Derivatives of the identity and generalizations of Milnor's invariants

We synthesize work of Koschorke on link maps and work of Johnson on the derivatives of the identity functor in homotopy theory. The result can be viewed in two ways: (1) as a generalization of Koschorkes ‘higher Hopf invariants’, which themselves can be viewed as a generalization of Milnors invariants of link maps in Euclidean space; and (2) as a stable range description, in terms of bordism, of the cross-effects of the identity functor in homotopy theory evaluated at spheres. We also show how our generalized Milnor invariants fit into the framework of a multivariable manifold calculus of functors, as developed by the author and Volic, which is itself a generalization of the single variable version due to Weiss and Goodwillie.

A manifold calculus approach to link maps and the linking number

We study the space of link maps Link(P1,...Pk;N), which is the space of maps P1 ` � � � ` Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation connects with recent work of the author, and is used to show that the Borromean rings are linked.

Multivariable manifold calculus of functors

Abstract. Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce “multivariable” manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. We construct multivariable Taylor approximations to such functors, classify multivariable homogeneous functors, apply this classification to compute the derivatives of a functor, and show what this gives for the space of link maps. We also relate Taylor approximations in single variable calculus to our multivariable ones.

CONFIGURATION SPACE INTEGRALS AND THE COHOMOLOGY OF THE SPACE OF HOMOTOPY STRING LINKS

Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in

A stable range description of the space of link maps

\mathbb{R}^n

Embeddings in the 3/4 range

for

Cubical homotopy theory

n>3

Cosimplicial models for spaces of links

since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of

Introduction to the manifold calculus of Goodwillie-Weiss ⁄

\mathbb{R}

Manifolds and -Theory

in