Embeddings in the 3/4 range

2011

Journal of Topology

Self Cite

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 Koschorke's ‘higher Hopf invariants’, which themselves can be viewed as a generalization of Milnor's 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 Volić, which is itself a generalization of the single variable version due to Weiss and Goodwillie.

“…A geometric understanding of this map is crucial to the main theorem of [11]. It is also easy to see how this is related to the linking number.…”

Section: Corollaries Of T Heorem 31mentioning

confidence: 98%

“…We begin with a very brief description of cobordism spaces. The author has used these in [11,12]. We review only the most basic details here.…”

Section: Cobordism Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Derivatives of the identity and generalizations of Milnor's invariants

2011

Journal of Topology

Self Cite

“…, P k ; N) for i = 1, 2. These spaces were used extensively in [21]. Identifying a good model for these spaces is necessary to define the maps in Theorems 5 and 6.…”

Section: Cobordism Spacesmentioning

confidence: 99%

A manifold calculus approach to link maps and the linking number

2008

Algebr. Geom. Topol.

We study the space of link maps Link(P 1 , . . . P k ; N ), which is the space of maps P 1 · · · P k → N such that the images of the P i 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.

“…Although statements and proofs here are usually combinatorially more complex, many definitions and techniques used in [26] carry over nicely to our multivariable setting. Manifold calculus has had many applications in the past decade [1,14,17,16,22,21,24]. With an eye toward extending some of them, we wish to generalize this theory to the setting where M breaks up as a disjoint union of manifolds, say M D`m i D1 P i .…”

Section: Introductionmentioning

confidence: 99%

Multivariable manifold calculus of functors

Volic

2012

Forum Mathematicum

Self Cite

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.