Title page for ETD etd-11152010-145059


Type of Document Dissertation
Author Tripathi, Girja Shanker
URN etd-11152010-145059
Title Orthogonal Grassmannians and Hermitian K-Theory in A^1-Homotopy Theory of Schemes
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Schlichting, Marco Committee Chair
Cohen, Daniel C. Committee Member
Delzell, Charles N. Committee Member
Hoffman, Jerome W. Committee Member
Sundar, Padmanabhan Committee Member
Srivastava, Ashok Dean's Representative
Keywords
  • Orthogonal Grassmannians
  • A^1-Homotopy Theory
  • Algebraic Geometry
  • Hermitian K-Theory
Date of Defense 2009-04-29
Availability unrestricted
Abstract
In this work we prove that the hermitian K-theory is geometrically representable in

the A^1 -homotopy category of smooth schemes over a field. We also study in detail a

realization functor from the A^1 -homotopy category of smooth schemes over the field

R of real numbers to the category of topological spaces. This functor is determined

by taking the real points of a smooth R-scheme. There is another realization functor

induced by taking the complex points with a similar description although we have

not discussed this other functor in this dissertation. Using these realization functors we

have concluded in brief the relation of hermitian K-theory of a smooth scheme over

the real numbers with the topological K-theory of the associated topological space

of the real and the complex points of that scheme: The realization of hermitian

K-theory induced taking the complex points is the topological K-theory of real

vector bundles of the topological space of complex points, whereas the realization

induced by taking the real points is a product of two copies of the topological

K-theory of real vector bundles of the topological space of real points.

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