POLISH ACADEMY OF SCIENCES WARSZAWA 200* SOME HOMOTOPY THEORETICAL QUESTIONS ARISING IN NIELSEN COINCIDENCE THEORY

Abstract. Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility prop-erties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems to be important for a more comprehensive understanding. Throughout this paper M and N will denote closed connected smooth manifolds of dimensions m and n ≥ 2, resp. Definition 1 (cf. [K2], (2), (3) and 1.1). Given (continuous) maps f1, f2: M → N, let MC(f1, f2) and MCC(f1, f2), resp., denote the minimum number of points and of path components, resp., among all the coincidence subspaces C(f ′1, f 2) = {x ∈M | f ′1(x) = f ′2(x)} ⊂ M of maps f ′i which are homotopic to fi, i = 1, 2. The pair (f1, f2) is called loose if MC(f1, f2) = 0 (or, equivalently, MCC(f1, f2) = 0), i.e. if the maps f1 and f2 can be deformed away from one another. The principal aim in topological coincidence theory is to get a good understanding of the minimum numbers MC(f1, f2) and MCC(f1, f2) (compare [B], p. 9) and, in particular, to obtain precise looseness criteria. In [K2] a geometric invariant ω#(f1, f2) was introduced: it is the bordism class of the (generic) coincidence submanifold (1) C = C(f1, f2) = {x ∈M | f1(x) = f2(x)