By J. F. Davis
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Additional info for A Survey of Spherical Space Form Problem
However rQ(8)x(g) = 0 and rZ/3x(g) = O. This completes our discussion for p-hyperelementary groups. Now we turn to a brief discussion of the finiteness obstructions for all P-groups. 19 Let 0 be a P-group of period n. (a) (T2"(0) =0. (b) (T"( 0) = 0 if 0 is of type I, IlK, Ill, IV(g), V, VI(g), or of type IV(b) or VI(b) with po= q (mod4). (c) Suppose 0 is not type VI(b) with p 0= 3 (mod 4). Then (T"( 0) = 0 if and only if (T"(H) = 0 for all 2-hyperelementary subgroups H. THEOREM Proof First assume 0 is of type I (metacyc1ic).
27 Let H n ( G; Z) = Z/IGI. Let X be apolarized (G, n -I, Z)-complex. Then X is finitely dominated, £leX) = x(k(X» E Ko(ZG)' Let X be an (n -I)-dimensional CW-complex with X = sn-l and fundamental group isomorphic to G. GI depends on the polarization, that is, on the choice of isomorphism of 1T j X with G and 1Tn -l(X) with Z. Two different k-invariants of X differ by an element of ±im(Aut G ~ Aut H n ( G; Z». Here Aut H n ( G; Z) = (Z/I GI)". Thus a classification of homotopy types of (n - 1)dimensional complexes X such that (a) 1T 1 X is isomorphic to G and (b) X= S n-I is given by elements of (Z/ G)'j (± I, im Aut( G».
16 (Swan ) Let gJ and g2 in Z/IGI represent additive generators of H n (G; Z) = Z/ IGI for a p-group G, then (a) X(gl)-X(g2)=[P g,'g,]E Tc (b) X(gJ u gJ ... 1 with the C j all free and finitely generated. (3) u n( G) vanishes if and only if there is a finite CWcomplex X n-I = S n-Ion which G. acts freely. COROLLARY The proof of (3) depends on the Milnor-Swan construction given in . 18 Un ( G) is the Swan finiteness obstruction, while the X(g) are the Wall finiteness obstructions for G. We also have the naturality properties for He G, rH(x(g)) = x(rH(g)) rH(un( G)) = un(H).
A Survey of Spherical Space Form Problem by J. F. Davis