Volume 74, Nº 3 (2019)

Conway's topographic approach to binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This leads naturally to a new class of commutative two-valued groups, which we call involutive. It is shown that the two-valued group of Conway's lax vectors plays a special role in this class. The group $\mathrm{PGL}_2(\mathbb Z)$ describing the symmetries of the Conway topograph acts by automorphisms of this two-valued group. Binary quadratic forms are interpreted as primitive elements of the Hopf 2-algebra of functions on the Conway group. This fact is used to construct an explicit embedding of the Conway two-valued group into $\mathbb R$ and thus to introduce a total group ordering on it. The two-valued algebraic involutive groups with symmetric multiplication law are classified, and it is shown that they are all obtained by the coset construction from the addition law on elliptic curves. In particular, this explains the special role of Mordell's modification of the Markov equation and reveals its connection with two-valued groups in $K$-theory. The survey concludes with a discussion of the role of two-valued groups and the group $\mathrm{PGL}_2(\mathbb Z)$ in the context of integrability in multivalued dynamics.Bibliography: 104 titles.

The first part of this survey gives a modernised exposition of the structure of the special unitary bordism ring, by combining the classical geometric methods of Conner–Floyd, Wall, and Stong with the Adams–Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 paper of Novikov. In the second part toric topology is used to describe geometric representatives in $SU$-bordism classes, including toric, quasi-toric, and Calabi–Yau manifolds.Bibliography: 56 titles.