By Andrew Ranicki

This booklet is an advent to surgical procedure thought: the traditional class process for high-dimensional manifolds. it really is aimed toward graduate scholars, who've already had a easy topology direction, and might now prefer to comprehend the topology of high-dimensional manifolds. this article comprises entry-level bills of many of the necessities of either algebra and topology, together with easy homotopy and homology, Poincare duality, bundles, co-bordism, embeddings, immersions, Whitehead torsion, Poincare complexes, round fibrations and quadratic kinds and formations. whereas targeting the elemental mechanics of surgical procedure, this booklet contains many labored examples, valuable drawings for representation of the algebra and references for additional analyzing.

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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.