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

By J. F. Davis

ISBN-10: 3718602504

ISBN-13: 9783718602506

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E. a 1T 1 X-module). The surgery semicharacteristic [13] is n XI/2(X;A)= I (-I)i[Hi(X;A)]EL~n+I(A). 25 (Davis [13]) Let a: (M 2n + l , VM)~ (X, g) be a degree one normal map. Then the image of s( a) in h • • L 2n +I(A) is XldM; A) - XI/2(X; A) where M is the induced 1T-cover of M. REVIEW OF SURGERY THEORY 269 Thus in the odd-dimensional case the difference of semicharacteristics gives a normal bordism invariant defined without preliminary surgery. If X is the universal cover of X then im (s( a)) E L;n+1 (A) has a geometric interpretation.

15 all the surgery obstructions for these projections are determined using the cohomology class k in H*(G/O, Z/2) via the map g:X -i> G/O'. In particular if g*(k*) == 0 then the surgery obstruction is zero. At this point we note that since k* is primitive (2g )*( k*) == 0 in mod 2 cohomology, so the surgery problem corresponding to (2g) must have trivial surgery obstruction. Moreover after doing surgery the total space of the universal cover will be some homotopy sphere, and for these addition in the group of homotopy spheres r "-1 and addition of maps give the same result.

11 or IV, there is one maximal 2-hyperelementary subgroup. 19(c) can be extended to the case VI(b), P == 3 (mod 4). However here the period n is congruent to 4 modulo 8 and G contains sub-groups of type IlL. R. Lee [23] proved that such a group cannot act freely on a closed manifold M such that H*(M; Zj2) = H*(sn~l; Zj2). Thus, from the point of view of the space form problem we are interested in 0"2n( G), which is zero. SECTION 4 A Brief Review of Some Facts in Surgery Theory Let Eo be the classifying space for stable spherical fiberings [26, 30].

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

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