HOMOTOPY EQUIVALENCES ON NON-IRREDUCIBLE 3-MANIFOLDS

This study involves homotopy equivalences on non-irreducible 3-manifolds and how they relate to homotopy groups.

Suppose M = M(,1) # ... # M(,t) where each M(,i) is a compact, p('2)-irreducible, sufficiently large, 3 -manifold. The following theorems comprise the main results:

Theorem 1. If g is a homeomorphism of M homotopic to the identity relative the boundary of M, then g is isotopic to the identity relative the boundary of M.

Theorem 2. Suppose M is boundary irreducible and f is a homeomorphism of M such that f(VBAR)(,(PAR-DIFF)M (UNION) x(,0)) = id and f induces the identity automorphism on the fundamental group. Then f is isotopic to r (CCIRC) d (CCIRC) k (rel (PAR-DIFF)M (UNION) x(,0)) where r is a rotation about a sphere, d and k are homeomorphisms with support in a collar neighborhood of the (PAR-DIFF)M.

Denote by H('id) the set of homeomorphisms which induce the identity automorphisms on (pi)(,1)(M,x(,0)) and are the identity on the (PAR-DIFF)M. Let H('Inn) be the set of homeomorphisms which induce an inner automorphism on (pi)(,1)(M,x(,0)) and are the identity on the boundary. Using Theorem 2 we prove the following theorems.

Theorem 3. Suppose M is boundary irreducible with p-tori boundary components and q-Klein bottle boundary components. Then

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Theorem 4. Suppose M is boundary irreducible with p-tori boundary components and q-Klein bottle boundary components. Then

(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)

The rotations mentioned above play a significant role in the following question: Suppose g is a map of a 3-dimenstional manifold such that the p('th)-iterate g('p) is homotopic to the identity. Is g homotopic to a map whose p('th)-iterate is equal to the identity? J. Tollefson and G. Heil have shown {Top Vol. 17, 1978} that for compact irreducible sufficiently large 3-manifolds the answer depends on whether a particular obstruction

Obs (Z(,p) , (pi)(,1)(M) , (psi)(,g)) (epsilon) H('p)(Z(,p), Z((pi)(,1)(M))) vanishes. We show that for any nontrivial rotation r on a closed manifold M = M(,1) # ... # M(,t) where each M(,i) is a compact P('2)-irreducible, sufficiently large 3-manifold that r('2) is isotopic to the identity but r is not homotopic to an involution even though Obs (Z(,2), (pi)(,1)(M), (psi)(,g)) = 0 in this case.