Self-intersections and higher Hopf invariants

Abstract

IN THIS paper we show how the well known models for loop spaces of Boardman and Vogt [3], James [5], May [9], and Segal[ lo], can be viewed in a natural way as “Thorn spaces for immersions”. Thus homotopy classes of maps into these models correspond to bordism classes of immersed manifolds with certain extra structures. By considering the multiple points of such immersions we obtain operations in homotopy theory. Special cases are the generalised and higher Hopf invariants of James[6], the Hopf ladder of Boardman and Steer[2], and the cohomotopy operations of Snaith [ 121, and Segal[ I I]. In 01 we establish the connection between models of loop spaces and structured immersions. In 02 we describe the process of “taking k-tuple points” in what might be termed an “external Hopf invariant”, and set out its properties in Theorem 2.2. We then get the Segal and Snaith operations by composing with a suitable “forgetful” function. A similar procedure is followed in 03 where the James operations are described. We are grateful to the referee for many helpful comments.