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ON STABLE PARALLELIZABILITY OF LENS SPACES
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ON STABLE PARALLELIZABILITY OF LENS SPACES
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ON STABLE PARALLELIZABILITY OF LENS SPACES
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Dissertation
ON STABLE PARALLELIZABILITY OF LENS SPACES
1982
Overview
The main purpose of this thesis is to define a free cyclic group action of prime order on a homotopy sphere which bounds a parallelizable manifold in a concrete way, and to show an algebraic characterization of its orbit space, called a lens space, to be stably parallelizable. Some special cases of these lens spaces have been considered by P. Orlik who defined a cyclic group action on a homotopy sphere represented as a Brieskorn manifold. The author also described the tangent bundle of a lens space and computed the characteristic classes of the lens space, correcting some errors which occurred in P. Orlik's paper (Smooth homotopy lens spaces, Michigan Math. J., 16 (1969) 245-255). By using the algebraic characterization of a lens space to be stably parallelizable, it is shown that \"there exists a stably parallelizable (2n-1)-dimensional lens space if and only if n is less than the order of the cyclic group which defines the lens space\". This gives some interesting results. For example, there are many ways to define free Z(,5)-actions on S('5) which are not linear so that these lens spaces are not diffeomorphic to any 5-dimensional classical lens space L(5; q(,1), q(,2), q(,3)). On 7-dimensional homotopy spheres, the author showed the following result: For p (GREATERTHEQ) 5 prime, there is a Z(,p)-action on C('5) and representative (SIGMA)'s of all 28 diffeomorphism classes of 7-dimensional homotopy spheres as submanifolds of C('5) such that the restriction of the Z(,5)-action on each homotopy sphere (SIGMA) is well-defined, and all induced lens spaces (SIGMA)/X(,p) are stably parallelizable\". Furthermore, for n (GREATERTHEQ) 3 odd, p (GREATERTHEQ) n, the standard sphere or the Kervaire sphere of dimension 2n-1 admits a free Z(,p)-action so that its orbit space is stably parallelizable. Finally, the author showed two generations of his work: one comes from a generalized description of a homotopy sphere as an algebraic variety, the other deals with the case of the composite moduli.
