Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität. / Includes bibliographical references ((p. 179-187)).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

No description available.

The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another. Owing to the difficulty in computing higher co-homology operations, one is led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. / Science, Faculty of / Mathematics, Department of / Graduate

The graded cohomology groups of a cartesian product of two cellular spaces are expressible in terms of the cohomology groups of the factors. This relationship is given by the (split) short exact Runneth sequence. However the multiplicative structure on the cohomology of a cartesian product can in general not be derived by solely referring to the Runneth formula. In this thesis we explicitly exhibit the cup product structure on a cartesian product of two (standard) lens spaces. This result is obtained by analyzing the Runneth sequence and by making use of the particular geometry of the spaces involved. / Science, Faculty of / Mathematics, Department of / Graduate

The work of this thesis is to motivate the following: Statement: The Hodge conjecture holds for products of varieties Z = XxC where (i) X is smooth, projective of dimension 2m-l, (ii) C is a smooth curve. The basic setting of this thesis is depicted by the following where (i) k⁻¹ (t) = Zt = Xt xC, {Xt } a Lefschetz pencil of hyperplane sections of X (ii) £ is the singular set of k, i.e., k = k is smooth and proper. Corresponding to this diagram are the extended Hodge bundle U H (Z . C) with integrable connection V , and the family of tt?1 t intermediate Jacobians. U JCZ ) with corresponding normal functions Now V induces an operator (also denoted by V) on the normal functions, and those normal functions v satisfying the differential equation Vv = 0 are labeled horizontal, which includes those normal functions arising from the primitive algebraic cocycles in H²m (Z). Now the known generalization of Lefschetz's techniques state that every primitive integral class of type (m,m) in H²m (Z) comes from a horizontal normal function in some natural way, so that what's needed to prove the above statement is some way of converting a normal function to an algebraic cocycle. We motivate this statement by proving some results about the group of normal functions, in particular our main result: Theorem: The group of normal functions are horizontal. To prove this theorem, we exhibit Vv as a global section of some holomorphic vector bundle over p¹, and then show that there are no non-zero global sections of this vector bundle. The main idea is to compare the quasi-canonical extensions of certain holomorphic vector bundles with integrable connection with those extensions arising from algebra (hypercohomology), by calculating certain periods of growth. Once this comparison is made precise, we apply a vanishing theorem statement about the global sections of the algebraic extensions to our geometric extensions, thus concluding the proof of the theorem. / Science, Faculty of / Mathematics, Department of / Graduate

The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.