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Download Group Actions on Manifolds by Schultz R. (ed.) PDF

By Schultz R. (ed.)

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Ik m+1 . k) be regular values such that 0 = r0 < c1 < r1 < c2 < . . < rk−1 < ck < rk = 1 . 20 (i) each (Wj ; Mj−1 , Mj ) = f −1 ([rj−1 , rj ]; {rj−1 }, {rj }) (1 j k) is an elementary cobordism of index ij . 18). Thus surgeryequivalent manifolds are cobordant. Conversely, note that every elementary cobordism is the trace of a surgery, and that by (i) every cobordism (W ; M, M ) is a union of elementary cobordisms. ✷ 0 If (W ; M, M ) has a Morse function f : W → I with critical points of index i0 i1 .

W Di Dm i+1 W0 (ii) An elementary (m + 1)-dimensional cobordism of index i is the cobordism (W ; M, M ) obtained from M × I by attaching an i-handle at S i−1 × Dm−i+1 → M × {1} , with W = M × I ∪ Di × Dm−i+1 .

W Di Dm i+1 W0 (ii) An elementary (m + 1)-dimensional cobordism of index i is the cobordism (W ; M, M ) obtained from M × I by attaching an i-handle at S i−1 × Dm−i+1 → M × {1} , with W = M × I ∪ Di × Dm−i+1 . (iii) The dual of an elementary (m + 1)-dimensional cobordism (W ; M, M ) of index i is the elementary (m + 1)-dimensional cobordism (W ; M , M ) of index (m − i + 1) obtained by reversing the ends, and regarding the i-handle attached to M × I as an (m − i + 1)-handle attached to M × I.

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