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4: In this proof ^ ( V rel x Q u (3V-F)) will be abbreviated by tfp and J(V rel x Q u (3V-F)) by J. 3 it was shown that #i/J is generated by spins and interchanges of 1-handles. When m=0 it follows that M ,/J =1. We 46 DARRYL McCULLOUGH AND ANDY MILLER now assume ra * 0. Thus $(#)/$(J) is generated by {[a ], [u> ] | i,j < m} where [ ] denotes the coset modulo $(J). ) are contained in J ) , and [a ] = [w ] - [co ] = [ax] show that ^(//p/^CJ) is generated by [o ]. 6 implies that Sl fi J so that [a ] t $(J) and $(^ 1)/$(J) = Z/2.

Let Gi,G2»•••fG^ be the components of 3V-F. 3 Proof: • #(M,F) is an isomorphism. 2. is isotopic to 1 M . Choose basepoints y Suppose f e Homeo(M,V) and f e F and y^ £ G. for l

Y) = P kj (y)P ij (x)u. k (x) 47. ) - P j k ( a j ) u l k ( x ) P l j ( x ) 48. 3 (a) right where y(x) e Aut(G ± ) is gx. Remarks In [F-R], the automorphisms are converted to operators acting on the on Aut(G). Hence our relations have the order of the elements reversed, compared to those in the list in [F-R]. (b) The errors. (c) In relations (34) in [F-R] contain typographical The corrected versions appear in our list as 18, 31, and 47. ) = relations (16), (23), and automorphism X.. (x) xa.. This accounts (called Y .

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