Dual Homotopy Invariants of G-Foliations

Hurder, Steven Edmond

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Description

Title

Dual Homotopy Invariants of G-Foliations

Author(s)

Hurder, Steven Edmond

Issue Date

1980

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

• This thesis defines and studies a new set of invariants for a foliation on a manifold M, where the normal bundle of is assumed to have a parallel G-structure. As an application of the theory which is constructed, it is shown that many of the secondary characteristic classes of smooth foliations are independently variable, extending the work of Heitsch in this direction. The homotopy groups (pi)(,n)(B(GAMMA)('q)) of the corresponding classifying space are shown to map onto (//R)('v(,n)), where {v(,n)} has a subsequence tending to infinity.
• For the group G = SO(q), all of the indecomposable secondary classes of Riemannian foliations are shown to be independent, and to vary independently when possible. This is a consequence of the following result which is proved in this thesis: Theorem. The classifying map (upsilon):B(GAMMA)(,G) (--->) BG of the normal G-structure on the classifying space of G-foliations is q-connected. Combined with the results of Lazarov-Pasternack, this theorem implies the stated results on the secondary classes. We also conclude that the homotopy groups (pi)(,n)(B(GAMMA)(,SO(q))) map onto (//R)('v(,n)), where v(,n) tends to infinity with n.
• For integrable complex foliations, it is shown that over half of the secondary classes are independently variable, extending the result of Baum and Bott on these classes. Surjections of (pi)(,n)(B(GAMMA)(,(//C))('n)) onto (//C)('v(,n)) are also shown to exist, where v(,n) tends to infinity with n.
• Finally, a geometric criterion is established for when the secondary classes of a G-foliation must be rational. For a general foliation of codimension q, this criterion is that the foliation be defined by a transverse map into a foliated manifold of dimension less than or equal 2q. For a Riemannian or complex foliation, it is required that the foliation be defined by a submersion.
• The invariants introduced in this thesis are constructed using Sullivan's theory of minimal models. By the Bott Vanishing Theorem, the Chern-Weil homomorphism h of the normal G-structure of the foliation on M gives a map h:I(G)(, ) (--->) (OMEGA)(M). Here, I(G)(, ) denotes the graded ring of Ad G-invariant polynomials on modulo the ideal of elements having degree greater than 2 , and is an integer depending only on G. We can always assume that (LESSTHEQ) q. It is shown that the algebra homotopy class of h is a well-defined invariant of the concordance class of . The induced map on dual homotopy, h('#):(pi)('*)(I(G)(, )) (--->) (pi)('*)(M), defines the dual homotopy characteristic invariants of . The structure of the space of invariants (pi)('*)(I(G)(, )) is given, and certain elements in this space are shown to correspond to the secondary characteristic classes of .
• The constructions used are all natural, so there is a universal map h('#):(pi)('*)(I(G)(, )) (--->) (pi)('*)(B(GAMMA)(,G)). This universal map is shown to be non-trivial for the groups G which are considered as a consequence of various results in the literature. A functional cup product operation is introduced, operating on the classes in (pi)('*)(I(G)(, )) by the generators of the algebra I(G) . This enables us to extend the known non-triviality of h('#). Relating this non-triviality back to the secondary classes gives our stated results.

Graduation Semester

1980

Type of Resource

text

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