##
Immersions and Mod-2 Quadratic Forms

by Louis H. Kauffman and Thomas F. Banchoff

This paper discusses the relationship between immersions and
mod-2 quadratic forms. This is of considerable interest not
only because there is a strong connection between the problems,
but also because the corresponding algebra in accessible and relevant.

This paper begins by discussing topology and how it leads to quadratic
forms. Quadratic forms are then examined in greater detail. In the
process, geometric proofs and interpretations apropos to the theory
mod-2 quadratic forms are developed.

Following, surfaces and their immersions are discussed, and an invariant,
B(f), for immersions of surfaces is introduced. The next section
provides examples of basic *homotopies* (handle sliding
and permutation), which are later shown to correspond to isomorphisms
of quadratic forms. This facilitates the classification of mod-2
forms, and the classification of immersion of surfaces
up to image homotropy can then be completed.