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An Introduction to Algebraic Surgery Andrew Ranicki

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An Introduction to Algebraic Surgery Andrew Ranicki An introduction to algebraic surgery Andrew Ranicki Introduction Surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory, without losing sight of the geometric motivation. 0.1 Historical background A closed m-dimensional topological manifold M has Poincar¶eduality isomorphisms m¡¤ » H (M) = H¤(M) : In order for a space X to be homotopy equivalent to an m-dimensional manifold it is thus necessary (but not in general su±cient) for X to be m¡¤ » an m-dimensional Poincar¶eduality space, with H (X) = H¤(X). The topological structure set ST OP (X) is de¯ned to be the set of equivalence classes of pairs (m-dimensional manifold M; homotopy equivalence h : M ! X) subject to the equivalence relation (M; h) » (M 0; h0) if there exists a homeomorphism f : M ! M 0 such that h0f ' h : M ! X: The basic problem of surgery theory is to decide if a Poincar¶ecomplex X is homotopy equivalent to a manifold (i.e. if ST OP (X) is non-empty), and if so to compute ST OP (X) in terms of the algebraic topology of X. Surgery theory was ¯rst developed for di®erentiable manifolds, and then extended to PL and topological manifolds. The classic Browder{Novikov{Sullivan{Wall obstruction theory for de- ciding if a Poincar¶ecomplex X is homotopy equivalent to a manifold has two stages : 82 Andrew Ranicki (i) the primary topological K-theory obstruction ºX 2 [X; B(G=T OP )] to a T OP reduction ºeX : X ! BT OP of the Spivak normal ¯bration ºX : X ! BG, which vanishes if and only if there exists a manifold M with a normal map (f; b): M ! X, that is a degree 1 map f : M ! X with a bundle map b : ºM ! ºeX , (ii) a secondary algebraic L-theory obstruction σ¤(f; b) 2 Lm(Z[¼1(X)]) in the surgery obstruction group of Wall [29], which is de¯ned if the obstruction in (i) vanishes, and which depends on the choice of T OP reduction ºeX , or equivalently on the bordism class of the normal map (f; b): M ! X. The surgery obstruction is such that σ¤(f; b) = 0 if (and for m ¸ 5 only if) (f; b) is normal bordant to a homotopy equivalence. There exists a T OP reduction ºeX of ºX for which the corresponding normal map (f; b): M ! X has zero surgery obstruction if (and for m ¸ 5 only if) the structure set ST OP (X) is non-empty. A relative version of the theory gives a two-stage obstruction for deciding if a homotopy equivalence M ! X from a manifold M is homotopic to a homeomorphism, which is traditionally formulated as the surgery exact sequence T OP ::: ! Lm+1(Z[¼1(X)]) ! S (X) ! [X; G=T OP ] ! Lm(Z[¼1(X)]) : See the paper by Browder [2] for an account of the original Sullivan-Wall surgery exact sequence in the di®erentiable category in the case when X has the homotopy type of a di®erentiable manifold O ::: ! Lm+1(Z[¼1(X)]) ! S (X) ! [X; G=O] ! Lm(Z[¼1(X)]) : The algebraic L-groups L¤(¤) of a ring with involution ¤ are de¯ned using quadratic forms over ¤ and their automorphisms, and are 4-periodic Lm(¤) = Lm+4(¤) : The surgery classi¯cation of exotic spheres of Kervaire and Milnor [7] in- cluded the ¯rst computation of the L-groups, namely 8 Z if m ´ 0 (mod 4) > < 0 if m ´ 1 (mod 4) L (Z) = m > Z if m ´ 2 (mod 4) :> 2 0 if m ´ 3 (mod 4) . The relationship between topological and PL manifolds was investigated using surgery methods in the 1960's by Novikov, Casson, Sullivan, Kirby and Siebenmann [8] (cf. Ranicki [23]), culminating in a disproof of the manifold Hauptvermutung : there exist homeomorphisms of PL manifolds which are not homotopic to PL homeomorphisms, and in fact there exist An introduction to algebraic surgery 83 topological manifolds without PL structure. The surgery exact sequence for the PL manifold structure set SPL(M) for a PL manifold M was re- lated to the exact sequence for ST OP (M) by a commutative braid of exact sequences $ $ 3 H (M; Z2) [M; G=P L] Lm(Z[¼1(M)]) MM q8 MMM qq8 MMM qqq MM qq M& qq MM& qqq SPL(M) [M; G=T OP ] q8 MM q8 MM qq MMM qqq MMM qqq M& qq M& L (Z[¼ (M)]) ST OP (M) H4(M; Z ) m+1 1 : : 2 with ¼¤(G=T OP ) = L¤(Z) : Quinn [17] gave a geometric construction of a spectrum of simplicial sets for any group ¼ L²(Z[¼]) = fLn(Z[¼]) j ­Ln(Z[¼]) ' Ln+1(Z[¼])g with homotopy groups ¼n(L²(Z[¼])) = ¼n+k(L¡k(Z[¼])) = Ln(Z[¼]) ; and L0(Z) ' L0(Z) £ G=T OP : The construction included an assembly map A : H¤(X; L²(Z)) ! L¤(Z[¼1(X)]) and for a manifold X the surgery obstruction function is given by A » [X; G=T OP ] ½ [X; L0(Z) £ G=T OP ] = Hn(X; L²(Z)) ! Lm(Z[¼1(X)]) : The surgery classifying spectra L²(¤) and the assembly map A were con- structed algebraically in Ranicki [22] for any ring with involution ¤, using quadratic Poincar¶ecomplex n-ads over ¤. The spectrum L²(Z) is appropri- ate for the surgery classi¯cation of homology manifold structures (Bryant, Ferry, Mio and Weinberger [3]); for topological manifolds it is necessary to work with the 1-connective spectrum L² = L²(Z)h1i, such that Ln is n-connected with L0 ' G=T OP . The relative homotopy groups of the spectrum-level assembly map Sm(X) = ¼m(A : X+ ^ L² ! L²(Z[¼1(X)])) 84 Andrew Ranicki ¯t into the algebraic surgery exact sequence ::: ! Lm+1(Z[¼1(X)]) ! Sm+1(X) A ! Hm(X; L²) ¡! Lm(Z[¼1(X)]) ! :::: The algebraic surgery theory of Ranicki [20], [22] provided one-stage ob- structions : (i) An m-dimensional Poincar¶eduality space X has a total surgery ob- struction s(X) 2 Sm(X) such that s(X) = 0 if (and for m ¸ 5 only if) X is homotopy equivalent to a manifold. (ii) A homotopy equivalence of m-dimensional manifolds h : M 0 ! M has a total surgery obstruction s(h) 2 Sm+1(M) such that s(h) = 0 if (and for m ¸ 5 only if) h is homotopic to a homeomorphism. Moreover, if X is an m-dimensional manifold and m ¸ 5 the geometric surgery exact sequence is isomorphic to the algebraic surgery exact se- quence T OP ::: / Lm+1(Z[¼1(X)]) / S (X) / [X; G=T OP ] / Lm(Z[¼1(X)]) =» =» ² ² A ::: / Lm+1(Z[¼1(X)]) / Sm+1(X) / Hm(X; L²) / Lm(Z[¼1(X)]) with T OP S (X) ! Sm+1(X);(M; h : M ! X) 7¡! s(h) : Given a normal map (f; b): M m ! X it is possible to kill an element x 2 ¼r(f) by surgery if and only if x can be represented by an embedding Sr¡1 £ Dn¡r+1 ,! M with a null-homotopy in X, in which case the trace of the surgery is a normal bordism ((g; c); (f; b); (f 0; b0)) : (N; M; M 0) ! X £ ([0; 1]; f0g; f1g) with N m+1 = M £ I [ Dr £ Dm¡r+1 ; M 0m = (MnSr¡1 £ Dm¡r+1) [ Dr £ Sm¡r : The normal map (f 0; b0): M 0 ! X is the geometric e®ect of the surgery on (f; b). Surgery theory investigates the extent to which a normal map can be made bordant to a homotopy equivalence by killing as much of ¼¤(f) as pos- sible. The original de¯nition of the non-simply-connected surgery obstruc- tion σ¤(f; b) 2 Lm(Z[¼1(X)]) (Wall [29]) was obtained after preliminary surgeries below the middle dimension, to kill the relative homotopy groups An introduction to algebraic surgery 85 ¼r(f) for 2r · m. It could thus be assumed that (f; b): M ! X is [m=2]- connected, with ¼r(f) = 0 for 2r · m, and σ¤(f; b) was de¯ned using the Poincar¶eduality structure on the middle-dimensional homotopy kernel(s). The surgery obstruction theory is much easier in the even-dimensional case m = 2n when ¼r(f) can be non-zero at most for r = m + 1 than in the odd-dimensional case m = 2n+1 when ¼r(f) can be non-zero for r = m+1 and r = m + 2. Wall [29,x18G] asked for a chain complex formulation of surgery, in which the obstruction groups Lm(¤) would appear as the cobordism groups of chain complexes with m-dimensional quadratic Poincar¶eduality, by anal- ogy with the cobordism groups of manifolds ­¤. Mishchenko [15] initiated such a theory of \m-dimensional symmetric Poincar¶ecomplexes" (C; Á) with C an m-dimensional f. g. free ¤-module chain complex d d d C : Cm ¡¡!Cm¡1 ¡¡!Cm¡2 ! ::: ! C1 ¡¡!C0 and Á a quadratic structure inducing m-dimensional Poincar¶eduality iso- ¤ m morphisms Á0 : H (C) ! Hm¡¤(C). The cobordism groups L (¤) (which are covariant in ¤) are such that for any m-dimensional geometric Poincar¶e complex X there is de¯ned a symmetric signature invariant ¤ m σ (X) = (C(Xe);Á) 2 L (Z[¼1(X)]) : The corresponding quadratic theory was developed in Ranicki [19]; the m-dimensional quadratic L-groups Lm(¤) for any m ¸ 0 were obtained as the groups of equivalence classes of \m-dimensional quadratic Poincar¶e complexes" (C; Ã).
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