Homotopy groups of PL-embedding spaces

Lawrence Stanislaus Husch, Jr.

Vol. 33 (1970), No. 1, 149–155

DOI: 10.2140/pjm.1970.33.149

Abstract

Let N be a compact PL-n-manifold, and let M be a PL-m-manifold without boundary. Two of the major problems in PL-topology are to determine conditions such that (1) any continuous map of N into M can be homotoped to a PL-embedding, and (2) two homotopic PL-embeddings are PL-isotopic.

If C(N,M) is the space of continuous maps of N into M with the compact open topology, and if PL(N,M) is the subspace of PL-embeddings, one can consider the map i_#;Π₀(PL(N,M)) → Π₀(C(N,M)) induced by inclusion. If (1) is true, then i_# is onto; if (2) is true, then i_# is one-to-one. In this paper, we investigate the higher homotopy groups of PL(N,M) and C(N,M).

Mathematical Subject Classification

Primary: 57.01

Milestones

Received: 11 April 1969

Published: 1 April 1970