A group-theoretic interpretation of Tutte's homotopy theory

Abstract

The Tutte-group T M of some given combinatorial geometry (or matroid) M, defined on a finite set E, has been introduced by A. W. M. Dress and W. Wenzel [ Adv. in Math. 77 (1989), 1–36] . T M is a finitely generated abelian group. In this paper the structure of a certain factor group T M of T M , called the truncated Tutte-group, will be determined completely. More precisely, it will be shown that Tutte's homotopy theory for matroids can be interpreted rather naturally as a systematic study of the relations between T M and the truncated Tutte-groups of the minors of M. This interpretation yields not only a complete knowledge of the structure of T M but also a very close relationship between this structure and the “ternarity” of M, leading to a new and essentially algebraic proof of the classical characterization of ternary matroids in terms of excluded minors, established originally by R. E. Bixby [ J. Combin. Theory B 26 (1979), 174–204] and P. D. Seymour [ J. Combin. Theory B 26 (1979), 159–173] . Furthermore, some partial results about T M itself are included.