1. Atlas: Regensburg Surgery Blockseminar 2012 The (smooth) surgery exact (Christoph Winges)

Definition 1.1. Let X be an n-dimensional Poincar´e complex, Mi closed n- , fi : Mi → X simple equivalences, i = 0, 1. We call f0 and f1 equivalent if there is a simple homotopy equivalence F (W, ∂0W, ∂1W ) −→ (X × I,X × {0},X × {1}) − 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) := {homotopy classes of simple homotopy equivalences}. Then s s E (X) acts on Sn (X) via composition. The obvious forgetful map s s Sn (X) → F (X) := {diffeomorphism classes of manifolds simply homotopy equivalent to X}

s s s factors to give a bijection E (X)\Sn (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 × {0}, ∂X × I ∪ X × {1})

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 → {±1} 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] ∈ 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 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 h|TM×{1} 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 ∈ N n(X)

 f  M / X Define η[f] := [f, f].

Definition of ∂: We define a action Ln+1(Zπ, w) y Sn(X): ' Let x ∈ Ln+1(Zπ, w), [f : M −→ X] ∈ Sn(X). Realize x by a normal map F :(W, ∂0W, ∂1W ) → (M × I,M × {0},M × {1}) 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 ∈ 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)] ∈ 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)])) = ∗ ∈ Sn(X). Now let x ∈ 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 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 × {0},S × {1})] ∈ 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 × {0, 1} . Similarly, the map f extends to a map n n+1 ∼ n+1 f ∪∂f idDn+1×{0,1} : N → S × I ∪Sn×{0,1} D × {0, 1} = S . We get a normal map 0 0 n+1 n+1 [(f , f )]: N → S ] ∈ 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 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]).