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.

Derivatives of the identity and generalizations of Milnor's invariants

“…The example above suggests a more general result when there are more than two components. The classification of multivariable homogeneous functors [13,Theorem 5.18], together with Proposition 6.7 above, implies the following lemma. Lemma 6.12.…”

“…Our generalization builds on Koschorke's ideas in the manner indicated in Section 1. It has been constructed from the point of view of a multivariable manifold calculus of functors, and so this requires a brief discussion of 'manifold calculus' (due to Weiss [15], Weiss and Goodwillie [7]) and a multivariable generalization of it (due to the author and Volić [13]). Calculus provides a natural organizational framework for these higher-order invariants.…”

“…The author and Volić [13] have developed a multivariable version of manifold calculus, motivated by the space of link maps. Throughout the rest of this section, we work with smooth closed manifolds P 1 , .…”

“…and it is enough by [13,Theorem 4.14] to check that it is an equivalence in the cases where the P i are either disks or empty, as Λ (1,1) is polynomial of degree at most (1, 1). If P 1 is a disk and P 2 = ∅, then the diagram in equation (9) reduces to Map(P 1 , N) Link(P 1 , ∅; N ).…”

“…The multivariable manifold calculus of functors is a generalization due to the author and Volić [13] of the manifold calculus of functors due to Weiss [15], Weiss and Goodwillie [7], built with the space of link maps in mind. What Corollary 1.2 tells us is that there is a stable range description, in terms of bordism, of certain layers in the multivariable manifold calculus tower of link maps, and that this stable range description, that is, the map H k , admits a geometric interpretation as a generalization of Koschorke and Milnor's invariants for link maps.…”