A Survey of Spherical Space Form Problem by J. F. Davis

By J. F. Davis

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Here it is possible for non-cyclotomic units to influence the Swan obstruction (at least if p == I (mod 8)). 15 [12] Let p be a prime not congruent to modulo 8. O ifp~ I (mod2 n-'). THEOREM In particular (T4( Q( 16,3, I)) ;£. O. This is a group of order 48 which gives the smallest possible group with a nonzero Swan obstruction. It is a subgroup of O~ for v> I. O if there is a uEZ[Az"·', Ap ] of norm However, we do not Know of any such examples when p == I (mod 8). -1. (T4( O~) ¥- 0 jor v> I. 6.

O. 6 L~(Z1TI (M», L~(Z1TI(M» which are algebraically defined, depend only on the fundamental groups and serve to measure the surgery obstructions in the non-simply connected case. 7 s:[(Xn,aXn),(G/( ),*)]~L~·S(Z1TI(X» which has image 0 = s( et), if and only if the corresponding surgery problem is normally cobordant to a homotopy equivalence of pairs (respectively s-cobordant to a simple homotopy equivalence of pairs), n 2: 5. 264 J. F. DAYIS AND R. J. MILGRAM If there is an Cl! 8 ~o(()(", a)("», Sfo (()(", a)("» of h-cobordism classes of degree 1 normal homotopy equivalences of pairs (fixed on a)(") and s-cobordism classes defined similarly.

There is a slight generalization of the above called Reidemeister-De Rham torsion. Suppose A ~ A is a mapping of rings. Now suppose E is a finite free complex over A, not necessarily based or acyclic. Then if A ® A E is acyclic we define where A®AE is given a basis induced by an A-basis of E. Let A be the kernel of the augmentation map Then and are well defined, with the class of [4J ] equal to 'T( D) - 'T( C). One can further show that 'T(D) is trivial and 'T( C) is in fact an invariant of the group G.

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