N n (S n ) L n (Z) L 5 (Z),

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1 . Maifold Atlas : Regesburg Surgery Blocksemiar 202 Exotic spheres (Sebastia Goette).. The surgery sequece for spheres. Recall the log exact surgery sequece for spheres from the previous talk, with L + (Z) S (S ) η N (S ) L (Z) L 5 (Z), S (S ) = {[f : Σ S ]... } /h-cob From the solutio of the Poicaré cojecture: All Σ above are homeomorphic to S. Theorem. (Hopf). All f : Σ S of degree oe are homotopic. This allows us to forget the map f i the defiitio of elemets of the structure set. Corollary.2 (from h-cobordism theorem). If 4, the S (S ) = { orieted diffeo types o S } =: Θ. With coected sum, Θ becomes a group, the group of exotic spheres i dimesio. The iverse of a exotic sphere is the same maifold with the opposite orietatio. Defiitio.3 ( [4, Defiitio 6.]). A almost framed -maifold (M, x 0, ū) is a -maifold M with a strog vector budle isomorphism ū: T (M \ {x 0 }) R a = R +a a almost framed bordism (W, γ, v) betwee (M i, x i, ū i ) for i = 0, cosists of a maifold W with W = M 0 M, a path γ : [0, ] W hittig i W trasversally i x i at t = i, ad a stable framig v of T (W \ im γ) that coicides with ū i o M i. Let the almost framed bordism group Ω alm be the set of classes with disjoit uio as additio. If we demad that the framigs exted over all of M or W, we aalogously get the framed bordism groups Ω fr. Propositio.4 ( [4, Lemma 6.9]). There is a atural isomorphism N (S ) = Ω alm Proof. Let f : (M, T M R a ) (S, ξ) be a smooth tagetial ormal map. Fix s S. After homotopy s is a regular value, ad f (s) = {x 0 } (because deg f = ). The budle data gives a stable almost framig because ξ S \{s} is trivial. Aalogously, bordisms of ormal maps ca be tured ito almost framed bordisms. For the iverse costructio, let ū: T (M \ {x 0 }) R a = R +a be a almost framig of M. Assume that M carries a Riemaia metric ad that ε is very small. By cotractig M \D ε (x 0 ) to a poit, we costruct a map f : M S with x 0 s. The almost framig ū iduces a fibrewise isomorphism f : T M R a ξ that is uique up to isomorphism. Similarly, a almost framed bordism gives rise to a bordism of ormal maps. The log exact sequece from above becomes... + L + (Z) Θ L (Z)... Defiitio.5. If a exotic sphere bouds a framed maifold we say Σ bp + (boudary of somethig parellelizable i dimesio + ). Note that bp + Θ is a subgroup uder coected sum.

2 2 Propositio.6 ( [4, Lemma 6.6]). We have bp + = ker ( η : Θ Ω alm Proof. If Σ ker η Θ, the there exists a bordism (F, F ): (W, T W R b ) (S, η) from the eutral elemet ). (id, f 0 ): (S, T S R) (S, R + ) to the elemet (f, f): (Σ, T Σ R a ) (S, ξ) of N (S ). The isomorphism v 0 : η = R ++b trivialises η, so we may assume that η = R ++b. Cosider the maifold N = W 0W D + ad exted F to T N R b R ++b. The F becomes a stable framig of N, so Σ = N bp +. Coversely, let Σ = N bp +, ad let f : Σ S be a smooth map of degree oe. The f ca be exteded to a smooth map F : N D + of degree oe, ad we may assume that 0 is a regular value of F with F (0) = {x} a sigle poit. For small ε > 0, the preimage F (D ε (0)) is agai a disk. Hece we obtai a bordism W = N \ F (D ε (0)) with 0 W = S N, W = N = M, ad with a map F : W D + \ D ε (0) S. The stable framig of N gives rise to a map F : T N R b R ++b, so we have costructed a ormal bordism. This shows that bp + ker η. () (2) (3) The log exact surgery sequece ow splits ito tractable pieces, 0 Θ η 0 Θ 2 η 2 sg Z Arf Z/2Z bp 0 for k 2, bp 2 0 for k 2, ad 0 bp 2j Θ 2j η 2j 0 for j Aother exact sequece. Recall the framed bordism groups Ω fr of Defiitio.3. The Potrijagi-Thom costructio gives a atural isomorphism Ω fr = π s = colim b π +b (S b ). Give a map α: S O(b), we get a family of rotatios of S b. Evaluatio gives a map S S b S b. This iduces a map f α : S +b = S S b ΣS b = S b, where deotes the joi ad Σ deotes ureduced suspesio. This is oe way to defie the J-homomorphism J : π (O) π s = Ω fr with J[α] = [f α ]. If M is almost framed, we ca represet T M R b as the pullback of a budle ξ S alog the collapse map f : M M/(M \ D ε (p)) = S. Stable vector budles over S are classified by π (BO), so we have a map : Ω fr π (BO). Fially, there is a atural forgetful map Ω fr Ω alm. Lemma.7. There is a log exact sequece (4) Proof. Exercise.... π + (BO) Ω fr }{{} =π s J π (BO)...

3 3.3. The sigature of almost framed bordism classes. Hirzebruch s sigature theorem says that sg(m) = L(T M)[M]. Let [M] Ω alm ad cosider the collapse map (f, f): (M, T M R b ) (S, ξ) i the defiitio of the map i (4). Because Potrijagi classes are atural ad f has degree oe, we coclude that sg(m) = L(ξ)[S ]. Now, the computatio of the possible values of sig(m) proceeds i two steps. () Idetify the image of Ω alm i KO 0 (S ) = π (O). (2) Determie L(ξ)[S ] for these vector budles. We use Lemma.7 to determie im = ker J. The groups π (O) = π + (BO) are periodic ad satisfy Z for 3, 7 mod, π (O) = Z/2Z for 0, mod, ad 0 otherwise. Regard the Beroulli umbers B Q with The first values are z e z = z 2 + ( ) + B (2)! t2. = B Theorem. (Adams, [, Theorems.,.3,.5]). If 3mod 4, the the J- homomorphism J : π (O) π s is ijective. Otherwise # ker J = deom B k. We come to step (2). Computatios with characteristic classes ad a little idex theory ca be used to show the followig. Theorem.9. A geerator ξ of KO 0 (S ) = π (BO) = π (O) satisfies L(ξ)[S ] = a k 2 2k (2 2k )B k 2k where a k = for k eve ad a k = 2 for k odd. Combie these two theorems to get Theorem.0. ad bp is cyclic. #bp = a k 2 2k 2 (2 2k ) um B k The first values are k #bp The value for k = is meaigless there are o exotic spheres i dimesio 3.

4 4 Proof. We have see that the map σ = sig factors over π (BO) = KO 0 (S ). From (), Ω 4 k alm sig Z bp 0 π (BO) L( )[S ] Theorem. computes im = ker J ad Theorem.9 computes the L-umber. Remark.. The Eells-Kuiper ivariat µ detects all elemets of bp 7 ad bp, see [2]. For Σ bp, oe either chooses N parallelisable with N = Σ ad takes a liear combiatio of sig(n) ad certai characteristic umbers of N. Or oe takes a liear combiatio of certai η-ivariats ad Cheeger-Cher-Simos correctio terms directly o Σ. For higher k, Stolz s s-ivariat also distiguishes all elemets of bp. It ca be defied ad computed i a similar maer [5]..4. The Kervaire-Ivariat-Oe-Problem. Because π 2 (BO) = 0, the forgetful map i (4) is surjective for = 2. Theorem.2 (Kervaire, Milor,..., Hill, Hopkis, Raveell [3]). The homomorphism Arf : Ω fr 2 = π s 2 Z 2 is surjective for 2 {2, 6, 4, 30, 62}, trivial for 2 / {2, 6, 4, 30, 62, 26}, ad ot yet kow for 2 = 26. Note: exceptio are of the form 2 j 2. Theorem.3. The group bp 2 are either trivial or Z 2 with { bp 2 0 if 2 {6, 4, 30, 62} = Z 2 if 2 {6, 4, 30, 62, 26} Proof. Use sequece (2)..5. The Cokerel of the J-homomorphism. We ow use the sequece (3) ad the left had sides of the sequeces () ad (2) to determie the remaiig groups Θ /bp +. For eve, bp + = 0, so Θ = Θ /bp +. By Theorem., the J-homomorphism J : π (O) Ω fr = π s is ijective except if 3 mod 4. Hece if 0 mod 4, sequece (4) becomes i particular, π (O) J Ω fr 0 = ker ( J : π (O) π ) s, Ω alm If =, we kow that ( sig ker : Ωalm = coker ( J : π (O) π s ). ) Z = ker ( : Ω alm π (O) ) because sig factors through the map L( )[S ]: π (BO) = π (O) Z, which is ijective by Theorem.9. As above, sequece (4) implies that ( sig ker : ( Ωalm Z) = coker J : π (O) π) s. Because the Arf ivariat is ot completely uderstood, we fially get the followig picture.

5 5 Theorem.4 ( [4, Theorem 6.46]). If {6, 4, 30, 62}, there is a short exact sequece () If / {6, 4, 30, 62, 26}, the (2) For = 26, either () or (2) holds. 0 Θ /bp + coker J Z 2 0. Θ /bp + = coker J. Corollary.5. All exotic spheres are stably parallelisable. Uless {6, 4, 30, 62, 26}, all framed bordism classes are realised by exotic spheres. It follows from Theorem. that coker J is closely related to the mysterious stable homotopy groups of spheres. The followig table gives the orders of the first few groups of exotic spheres. We do ot go ito their group structures here #bp #π s # coker J #(Θ /bp + ) #Θ Fat umbers either represet exceptios due to the Kervaire ivariat oe problem, see Theorems.3 ad.4, or they are due to π k+i (BO) = Z 2 for i {, 2}. Refereces [] J. F. Adams, O the groups J(X). IV, Topology 5 (966), [2] J. Eells, N. Kuiper, A ivariat for certai smooth maifolds, Aali di Math. 60 (962), [3] M. A. Hill, M. J. Hopkis, D. C. Raveel, O the o-existece of elemets of Kervaire ivariat oe, preprit (2009), arxiv: [4] W. Lück, A Basic Itroductio to Surgery Theory, preprit 2004, 2, 5 [5] S. Stolz, A ote o the bp-compoet of (4 )-dimesioal homotopy spheres, Proc. Amer. Math. Soc. 99 (97)