22 P. DEIFT, L. C. LI, AND C. TOMEI

which agrees with (2.52), verifying the homomorphism property. To show that $ is onto,

suppose ((Af, A ) , ( X , F ) ) £ Go, det Af, det X 0. We must show that there exists

g = g+gZl € G such that $(g) — ((M , A"), (X,Y)). Recall that we must have g(z) = g(z)

and g(-) homotopic to the identity. By the homomorphism property, and by the symmetry

between z = 0 and z = oo, it is enough to show that there exists g = g+(z) (necessarily

#+(l) = 7-(l) = I) such that

(g+(0)-\g+(0r1g'+(0))=(M,K) = (M,Q)o(I,K) , (2.53)

and again by the homomorphism property it is enough to consider (A/, 0) and (/, AT)

separately.

For (/, A'), a suitable loop is

g = g+ = e*(1~z)K , (2.54)

with homotopy

9t = (gth = etz{1-x K , 0 t 1 . (2.55)

For (A/, 0) with det M 0, it is sufficient by the homomorphism property and connectivity

to consider the case

M = ( J - h c ) - 1 (2.56)

where the real matrix e is small. But here

g = g+ = I + e(l-z2) (2.57)

is an appropriate loop, deformable to the identity via

9t = (gt)+=I + te(l-z2), 0 * 1 . (2.58)

This completes the proof that $ is surjective.