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If E is an arbitrary infinite-dimensional normed linear space, it is shown that if all homeomorphisms of E onto itself are stable, then the annulus conjecture is true for E. As a result, this confirms that the annulus conjecture for Hilbert space is true. A partial converse is that for those spaces E which have some hyperplane homeomorphic to E, if the annulus conjecture is true for E and if all homeomorphisms of E onto itself are isotopic to the identity, then all homeomorphisms of E onto itself are stable.doi:10.1090/s0002-9939-1970-0256419-6 fatcat:e4ha3gmrwjdzhpatqvktzv5m6e