Subject: Re: two questions From: Clarence W Wilkerson Jr Date: Tue, 05 Oct 2004 10:55:54 -0500 The embeddings of open disks in R^n: I have a vague memory of work of Ed Connell in the 60's on stable ( not the homotopy theory definition ) maps which was then incorporated in Kirby's work. I'll check MathSciNet to see if I can reconstruct this shadow in my wetware. Ok, here's what I was trying to remember.A homeo of R^n to itself is stable if it can be written as a finite composition of maps, each of which is a homeomorphism fixing some open set ( not necess. same open for each). The Stable Homeomorphism conjecture is that all orientation preserving homeomorphisms of R^n are stable. I believe this is related to the question asked, since each stable homeomorphism is isotopic to the identity homeo. That is, I'm claiming that Homeo(R^n,R^n)_orientationpreserving is pathconnected. if n \neq 4. References: MR0242165 (39 #3499) Kirby, Robion C. Stable homeomorphisms and the annulus conjecture. Ann. of Math. (2) 89 1969 575--582. 57.01 The remark under Theorem 1 says that the SHC is true except in dimension 4. Clarence