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1977 (Publication #25)

Immersions and Mod 2 Quadratic Forms (with Lou Kauffman)

American Mathematical Monthly 84 (1977), 168-185. (Awarded Lester Ford Award for exposition, Summer 1978).

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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.