| Summary: | The objects of study in this thesis are symmetric products and spaces of algebraic cycles. The first new result concerns symmetric products and it describes the geometry of truncated symmetric products (or, in other terminology, symmetric products modulo 2). We prove that if <I>M</I> is a closed compact connected triangulable manifold, a necessary and sufficient condition for its symmetric products modulo 2 to be manifolds is that <I>M</I> is a circle. We also show that the symmetric products of the circle modulo 2 are homeomorphic to real projective spaces and give an interpretation of this homeomorphism as a real topological analogue of Vieta's theorem. The second result concerns the spaces of real algebraic cycles, first studied by T.K. Lam. We describe a method of calculating the homotopy groups of the spaces of real cycles with integral coefficients on projective spaces; we give an explicit formula for the groups which lie in the "stable range". The third result (or, rather, a group of results) is the construction of a quaternionic analogue of Lawson's theory of algebraic cycles. We define quaternionic objects as those, which are invariant (in the case of varieties) or equivalent (in the case of polynomials) with respect to a free involution on CP<I><SUP>2n</SUP></I><SUP>+1</SUP>, induced by the action of the quaternion <I>j</I> on H<I><SUP>n</SUP></I>. Basic properties of quaternionic algebraic cycles are studied; a rational "quaternionic suspension theorem" is proved and the spaces of quaternionic cycles with rational coefficients on CP<I><SUP>2n</SUP></I><SUP>+1</SUP> are described. We also present a method of calculating the Betti numbers of the spaces of quaternionic cycles of degree 2 and odd codimension on CP<SUP>∞</SUP>. Some other results that are included in the thesis are a twisted version of the Dold-Thom theorem and an interpretation of the Kuiper-Massey theorem via symmetric products. After the main results on quaternionic cycles were proved, the author learned that similar results were obtained by Lawson, Lima-Filho and Michelson. Their version of the quaternionic suspension theorem is stronger and requires more sophisticated machinery for the proof.
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