1. Manifold Atlas: Regensburg Surgery Blockseminar 2012 The (smooth) surgery exact sequence (Christoph Winges) Definition 1.1. Let X be an n-dimensional Poincar´e complex, Mi closed n- manifolds, fi : Mi ! X simple homotopy equivalences, i = 0; 1. We call f0 and f1 equivalent if there is a simple homotopy equivalence F (W; @0W; @1W ) −! (X × I;X × f0g;X × f1g) − together with degree one diffeomorphisms g0 : M0 ! @0W , g1 : M1 ! @1W such that @iF ◦ gi = fi. The collection of equivalence classes of such maps is the simple structure set s Sn (X). If X is a manifold, the structure set has a canonical basepoint ∗ given by id: X ! X. Remark 1.2. (1) Note that any cobordism W as above is automatically an h-cobordism. So the s-cobordism theorem applies to show that the equiva- s lence relation for the Sn (X) reduces to the existence of a homotopy com- mutative triangle ∼= M0 / M1 zz f0 zz zz f1 z} z X , where the horizontal map is a degree one diffeomorphism. (2) Let E s(X) := fhomotopy classes of simple homotopy equivalencesg. Then s s E (X) acts on Sn (X) via composition. The obvious forgetful map s s Sn (X) ! F (X) := fdiffeomorphism classes of manifolds simply homotopy equivalent to Xg s s s factors to give a bijection E (X)nSn (X) ∼ F (X): Definition 1.3. Let X be an n-dimensional, compact manifold and (f; f): M ! X a normal map of degree one such that @f is a diffeomorphism. A nullbordism of (f; f) consists of • a degree one map of manifold triads F :(W; @0W; @1W ) ! (X × I;X × f0g; @X × I [ X × f1g) with @1F a diffeomorphism, • a degree one diffeomorphism g :(M; @M) ! (@0W; @0W \ @1W ), a • a bundle map F : TW ⊕ R ! η and a • an isomorphism ξ ⊕ R =∼ η satisfying appropriate compatibility conditions. The definition of a nullbordism allows us to derive the notion of a normal cobordism. The set of normal bordism classes is denoted Nn+1(X; @X). Theorem 1.4 (Browder-Novikov-Sullivan-Wall exact sequence). Let X be a smooth, connected, closed manifold, n ≥ 5. Set π := π1X and let w : π ! {±1g be the ori- entation homomorphism. Then the following is an exact sequence of pointed sets. σ @ η σ Nn+1(X × I;X × @I) / Ln+1(Zπ; w) / S (X) / Nn(X) / L n(Zπ; w) Notation. The map σ is the (simple) surgery obstruction. We give the defini- tions of η and @ for homotopy equivalences, the definitions in the simple case are analogous. 1 2 ' −1 Definition of η: Let [f : M −! X] 2 Sn(X): Let f be a homotopy inverse of ∗ f. Set ξ := f −1 TM. Pick a homotopy h: id ' f −1 ◦ f. There is a lift of h to a bundle map h: TM × I ! TM. The universal property of a pullback gives the dashed arrow in the following diagram: f ∗ξ / TM < y xx y =xx y xx y hjTM×{1g xx TM / TM M / M xx xx =xx =xx xx xx xx xx M / M f −1◦f By composing the dashed arrow with the natural map f ∗ξ ! ξ, we obtain a map f : TM ! ξ and thus f TM / ξ 2 N n(X) f M / X Define η[f] := [f; f]. Definition of @: We define a group action Ln+1(Zπ; w) y Sn(X): ' Let x 2 Ln+1(Zπ; w); [f : M −! X] 2 Sn(X). Realize x by a normal map F :(W; @0W; @1W ) ! (M × I;M × f0g;M × f1g) as in the version of the Wall realization theorem given in the previous talk. Then set x · [f] := [f ◦ @1F ]: Define @(x) = x · [id: X ! X]: Exactness. • Exactness at Nn(X): This is one of the main results that were established so far. • Exactness at Sn(X): Let x 2 Ln+1(Zπ; w). Realize x by a normal map F :(W; @W ) ! (X × I;XI). Then @1F is bordant to a diffeomorphism (as witnessed by W ), so η([@1F ]) = ∗. Suppose now that η([f : M ! X]) = ∗. Then η([f]) is normally bor- dant to a diffeomorphism. Pick any cobordism witnessing this and take its surgery obstruction to obtain a preimage of [f]. • Exactness at Ln+1(Zπ; w): Start with an element [(f; f)] 2 Nn+1(X × I;X × @I). Obviously, a realization of σ([(f; f)]) is given by (f; f) itself; since @f is a diffeomorphism, we have that @(σ([(f; f)])) = ∗ 2 Sn(X). Now let x 2 Ln+1(Zπ; w) be mapped to ∗ via @, i.e. if we realize x by a normal map F as in the Wall realization theorem, then @1F is h- cobordant to a diffeomorphism. Pick any homotopy equivalence F 0 that witnesses this (cf. the definition of the structure set), then glue F and F 0 00 00 along the common boundary @1F to get a normal map F . Now @F is a diffeomorphism, so we have defined an element in Nn+1(X × I;X × @I). Since the surgery obstruction behaves additively with respect to gluing, 3 and since an h-cobordism has surgery obstruction 0, the element [F 00] is a preimage of x. Remark 1.5. The statement about exactness at Sn(X) can be strengthened: Two elements in the structure set have the same image under η if and only if they lie in the same Ln+1(Zπ; w)-orbit. Example 1.6 (Homotopy spheres). Construct a map n n n+1 γ : Nn+1(S × I;S × @I) ! Nn+1(S ): Let n n n n n [(f; f):(M; @0M; @1M) ! (S × I;S × f0g;S × f1g)] 2 Nn+1(S × I;S × @I): Recall that @f is a diffeomorphism. This allows us to construct a closed manifold n+1 N := M [@f D × f0; 1g : Similarly, the map f extends to a map n n+1 ∼ n+1 f [@f idDn+1×{0;1g : N ! S × I [Sn×{0;1g D × f0; 1g = S : We get a normal map 0 0 n+1 n+1 [(f ; f )]: N ! S ] 2 Nn+1(S ) Proposition 1.7. γ is a bijection. After checking that the surgery obstruction maps n n n+1 σ : Nn+1(S × I;S × @I) ! Ln+1(Z[e])andσ : Nn+1(S ) ! Ln+1(Z[e]) agree under the identification provided by γ, this allows us to splice the various surgery exact sequences together to form one long exact sequence η ::: n+1 σ @ n n σ / Nn+1(S ) / Ln+1(Z[e]) / Sn(S ) / Nn(S ) / Ln(Z[e]):.