Higher-dimensional link operations and stable homotopy

Ulrich Koschorke and Dale Rolfsen

Vol. 139 (1989), No. 1, 87–106

DOI: 10.2140/pjm.1989.139.87

Abstract

The fundamental invariant, α, of link homotopy in higher dimensions takes values in the stable homotopy ring Π_∗^S. Using the Eckmann-Pontrjagin-Thom correspondence between Π_∗^S and Ω_∗^fr, the framed bordism ring, we give new methods of calculating α and its nonstable version, A, and an extended definition to link maps of arbitrary dimensions. Also we show that the set of all links of two components (arbitrary dimensions) has a natural ring-like structure, compatible with homotopy. The geometric approach allows us to show these operations are compatible, via A, with the ring structure of the homotopy groups of spheres and of Π_∗^S. Finally, this introduces a new bifiltration Π_n^p,q of Π_∗^S, which is of independent interest.

Mathematical Subject Classification 2000

Primary: 57R40

Secondary: 57Q45, 57R42

Milestones

Received: 15 July 1987

Revised: 18 May 1988

Published: 1 September 1989