By Max Deuring

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First we extend the Spin(4)-action on ^ ( H ) to all of (f2*(H), 6) using the isomorphism ClgH fin(H) =* fii(H) ® (H/Q)®^- 1 ). i)92)(«o) = p{Q2){ao) for a 0 € fi0(H), and o{q1,qi){uda2 .. dan) = a(g l 5 g 2 )Md(/o(gi)(a 2 )) .. d{p{qi){an)), for n > 1. Similarly we extend the involution r by T(a0) = ao for ao 6 fio(ff), r(aodai) = —(dai)do and for n > 2 r(wda 2 .. dan) = - ( - l ) 2 ( " - 2 H n - 3 ) r ( u ; ) d a n .. da 2 . With these definitions r commutes with the Spin(4)-action and the boundary map so that in particular Qn,In{S) and Bn(M) split into (±l)-eigenspaces for r as Spin(4)-modules.

9) Hg(C*(U)) = Z 0 q = 0 or p otherwise. , aq) in U gives rise to a singular ^-simplex fa : A 9 -» U and one shows (cf. 21) t h a t the inclusion C*{U) —¥ C* ing (f/) into the singular chain complex induces an isomorphism in homology. Let us write 0(U) = HP(U) = Z and call it the orientation module for U since an isometry g of U induces multiplication by ± 1 depending on g being orientation preserving or not. Thus 0(X) = Z* in our previous notation. Also we put O(0) = Z . Finally for U C X a p-dimensional subspace let f/1 denote the subspace perpendicular to U.

With these definitions r commutes with the Spin(4)-action and the boundary map so that in particular Qn,In{S) and Bn(M) split into (±l)-eigenspaces for r as Spin(4)-modules. The following lemma is straight forward. 10. i) Spin(4) acts trivially on HH*(M){= ii) On HHn(E) the involution T is T — (—l)"id. iii) For S2 C Spin (4) acting on fin(H) we have fi|). 10, i). 10, iii) for Chapter 6. Sydler's theorem and non-commutative differential forms 51 n — 2. ) except for the identification of the map J which follows by direct checking of the maps involved.

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