By Rutherford D.

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**Example text**

Thus its fixed point set is non-empty and con- If J = _ j - I then j2 = -I, so this time we are finding complex structures on V~ which commute with the action of H. )) preserves ~ 9on-singula~ innerproduct, structures J : (H,V) then the family o_ffcomplex > (H,V) for which (i) (v,J(w) = -(J(v),w) = (w,J(v)) (2) (v,J(v)) > O !! v ~ O forms a non-empty connected subset of GL(V). The relation of these J to the computation of ind (H,V ~ R C,<-,~) is seen by introducing D(v + iv') = J(v') - iJ(v) - Note that D 2 = I.

Thus there is y ~ 's~)('Z'p2 n ! (y) = mx. then ~ We s e t ! (my (Yr(y))/m Lc(x) : (Tr(y))/m. If#(y ! with ! ) = m x , - m y) = 0 so mTr(y ) = m Tr(y) or = (Tr(y))/m = Lc(x), which is well defined. * * p S s s-1 _>. _mm ) pS pS _ (Z s; 2n 2n ~ _ also "" - 46 - [6, sec. 7] we were able to compute~// . then ~ . Set / J = (pS-l)/2, / can be described as the suitably defined bordism classes of objects C~l,... , # ~ ) is an ordered#-tuple oriented manifold. of complex vector bundles over a closed Addition in ~ and the ring structure in ) is by disjoint union is given by > M2n][(#l,''', ~ 2 ) !

9 )). functions f : H We are Let W be the linear space of all > V which satisfy f(kh) = kf(h) for all k e K~ h e H. by (~f)(h) = f(h~). <'f,f ) = he H (f(h), (K, V, (. , 9 ) ) = e A representation The innerproduct (H~ W) is then given on W is given by f (h)). We set (H, V,<"" , "> ), which defines an additive HK homomorphism e~< : RI(K ) > RI(H ). - Choose D : (K~ V) 31 - > (K, V) such that (I) D2 = I (2) (v,Dv) = (Dv,w) (3) (v,Dv) > 0 if v / 0 we set D(f)(h) = D(f(h))~ then D satisfies (i) - (3) also.

### Substitional Analysis by Rutherford D.

by Thomas

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