A homotopy boundary of a finite simply connected CW complex $X$ is the boundary of a compact manifold of the homotopy type of $X$. We study the rational homotopy type of the homotopy boundaries of $X$ using Quillen models and we show, in particular, that a large class of homotopy boundaries of $X$ is rationally determined by the rational homotopy type of $X$.

Un bord homotopique d’un CW-complexe fini simplement connexe $X$ est, par définition, le bord d’une variété compacte du type d’homotopie de $X$. Nous étudions le type d’homotopie rationnelle des bords homotopiques de $X$ à l’aide des modèles de Quillen et nous montrons, en particulier, qu’une large classe de bords homotopiques de $X$ est rationnellement déterminée par le type d’homotopie rationnelle de $X$.

We introduce equivalence relations among asymptotic homomorphisms that in general are stronger than homotopy, but which we show are equivalent to homotopy when the domain is a suspended $C^*$-algebra. As an application, we show that the E-theory of Connes and Higson can be realized as a special case of Kasparov’s KK-theory.

We describe away of representing finite biquandles with $n$ elements as $2n \times 2n$ block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the nontriviality of the virtual trefoil and various Kishino knots. We also exhibit an oriented virtual knot which is distinguished from both its obverse and its reverse by a finite biquandle counting invariant. We classify biquandles of order 2, 3 and 4 and provide a URL for our Maple programs for computing with finite biquandles.

We provide a direct connection between the work of David Benson and Mark Feshbach, and the work of John Martino and Stewart Priddy on stable splittings of classifying spaces of finite groups.

In this thesis, we are concerned with the study of cohomology with local coefficients and applications to (non-orientable) real vector bundles. The thesis consists of an introduction and three separate sections. The introduction gives some motivation for considering cohomology with local coefficients and an outline of the results obtained. The first section deals with a general discussion of such cohomology groups and contains a Künneth Theorem for such groups. The second section is devoted to some computations which are needed later and the final section gives a complete description of the integral cohomology of the spaces $BO(n)$ and $BSO(n)$.

We develop the deformation theory of $A_\infty$ algebras together with $\infty$-inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, $A_\infty$ algebras, associative algebras with inner products, and $A_\infty$ algebras with inner products.