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1. Introduction. the Inertia Group I(M) of an Oriented Closed Smooth Manifold M Is Defined

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1. Introduction. the Inertia Group I(M) of an Oriented Closed Smooth Manifold M Is Defined J. Math. Soc. Japan Vol. 21, No. 1, 1969 On the inertia groups of homology tori By Katsuo KAWAKUBO (Received Feb. 15, 1968) § 1. Introduction. The inertia group I(M) of an oriented closed smooth manifold M is defined N to be the subgroup of E consisting of those homotopy spheres S which satisfy the condition MSS= M, where On is the group of homotopy n-spheres. This group I(M) is one of the diffeomorphy invariants of M. The inertia groups of manifolds have been studied by I. Tamura [14], C. T. C. Wall [18], S. P. Novikov [10], W. Browder [1] and A. Kosinski [6]. The following problems have been proposed by them as important ones : (I) Is it combinatorially (or topologically) invariant? (II) Does it depend on more than the tangential homotopy equivalence class at the manifold? (W. Browder cf. [7]) (III) Is it contained in 0(a~r), if we restrict the manifold within it- mani-folds?(S. P. Novikov [10]) In this paper the following facts will be proved which answer the problems above. The inertia groin of S 3X S 14 is not combinatorially (therefore not topologi- cally) invariant and depends on more than the tangential homotopy equivalence class of S 3 XS 14 For ,514 S14, I(S3xS14) contains a homotopy sphere S17 which does not be- long to 017(air). The author wishes to express his hearty thanks to Professor I. Tamura for suggesting the viewpoint of this research and for showing the attitude of mind in mathematics and also to Mr. H. Sato, Mr. M. Kato and Mr. T. Akiba for many invaluable discussions. § 2. Notations and results. In this paper all the manifolds are compact connected smooth oriented manifolds and the diffeomorphisms are orientation preserving. We write M1= M2 for manifolds M1, M2, if there is an orientation preserving diff eomor- phism f : M1-M2. Let Oq be the group of homotopy q-spheres and hq the pseudo-isotopy 38 K. KAWAKUBO group of diffeomorphism of S g-1. It is well known that Oq and I'q are equi- valent (q 3) (Smale [12]). A subgroup of q-dimensional homotopy spheres which bound parallelizable manifolds is denoted by Oq(a'r). The inertia group of a closed differentiable manifold Mn is defined to be the group {S E On MTh S = Mn} which is denoted by 1(M). Now we shall define pairings K1, K2: for 0 < p < q : K1: ;rp(SOq+i)X rq+1 i~ rp+q+l , K2: ~rp(SOq+I)x 7Cq(SOp+i) rp+q+l Let h E Tp(SOq+i),r E rq+1= 2ro(diffS q)/7r0(diffD+1). (In the following we will use the same symbol for an element of a group as its representative.) We define the diffeomorphism F: Spx Sq--Sp x Sg by F(x,y) = (x,rh(x)r~1(y)). Attaching two manifolds, W1= Dp~1 x S g and W2 = Sp x Dg`1, by the diffeomor- phism F : S1' x S q -~ Sp X S q, we have a manifold I=Dp~1xSgUSpxDq~1, F Making use of the Mayer-Vietoris exact sequence, it is easy to see that the manifold ~' is a homotopy sphere for p < q. We assume that the orientation of the manifold A U B compatible with the first part A, is given. The pairing .f K1 is defined by K1(h, r) = .7. We shall now prove that this does not depend on the choice of represent- atives. Let h' E 7cp(SOq+1)be other representative and H : S'1 X I - *SOq+1 be a homo- topy between h and h'. We write H(x, t) as hr(x). Let r' E I'q+1 be other re- presentative and R: S g x 1-~ S g x l be a pseudo isotopy between r and r'. Let p1: S g x I--~ S g and p : S g x I - I be the projections to the first and to the second respectively. We now define the diffeomorphism G: Sp x S gx I Sp x S g x I by G(x, y, t) = (x,R(ht(x)p1R~1(y, t),p2R-i(y, t))). Attaching two manifold D'11 X S q X I and Sp x Dg+1 x I by the dilleomorphism G: SpxSgXl-~SpxSgXl, we have a manifold X=Dp+iXSgxlUSpxDq+1XI. G The boundary aX is composed of the disjoint sum 27 and --27' where 27' is aa homotopy sphere made from h' and r'. Making use of the Mayer-Vietoris exact sequence, it is easy to see that inclusions i : 27 --f X and i' : 27' -->X give the homotopy equivalences. Therefore 27 is cliff eomorphic to 27' by the h-cobordism theorem. Since On = T n (n 3), 21 and 21' are the same element of rp+q+1. The inertia groups of homology tori 39 Next we define the pairing K2. Let h1 7rp(SOq+1),h2 7rq(SOp±1). We con- sider two bundles (B1, Sp+1, Dq+1, p1) and (B2, Sq+1, Dp+1, p2) with characteristic maps h1 and h2 respectively. We can write B1= Dp+l X Dq+l U Dp+1X Dq+1 and h•1 B2 = D+ 1X Dp+1 U D~+1X Dp+1 where D±+1x Dg~1= p~ 1(D+1) and D±+1X Dp-~1= p21(Dt+l). The plumbing manifold of B1 and B2 is defined to be the oriented differentiable (p+q+2)-manifold obtained as a quotient space of B1 U B2 by identifying p1(D1) = Dp+1 X Dq+l and p1(D1) = Dq+1X D1 by the relation (x, y) _ (y, x) (x E Di+1= Dp+1, y Dq 1= Dr') and is denoted by B1 V B2. The boundary a(B1 B2) can be seen as follows. Let f : S p X S'-~S Q X Sp bethe cliff eomorphism defined by f (x, y) _ (hi(x)y,h2(h1(x)y)x}. Attaching two manifold D++1X S q and Dq+1X S ~' by the diff eomorphism f: S+ X S 3_ S4 X S p, we have D++1X S g U D i 1 X SP = a(B1 B2). Therefore a(B1 B2) is a homotopy .f sphere by the same argument of the pairing K1. The pairing K2 is defined by K2(h1,h2) = a(B1 B2) and one can easily prove that it is well-defined like K1. I'p,q denotes the subgroup of rp+q_1 generated by the image of the pairing K2: 7rp_1(SOq)X7cq_1(SOp)--~Fp+q_1.I'p,Q denotes the subgroup of F,q generated by the image of the restricted pairing K2: 1Cp_1(SOq)Xs7cq_1(SOp_1)-~hp+q_1 where s is a natural map s : 7rq_1(SOp_1)-7rq_1(SOp). Then the following theorems will be proved. THEOREMA. Let Mm be the simply connected 7c-manifold with HfMm) = 0 for i ~ 0, p, q, p+q = m. Then I(Mm) C rp+l,q for p < q-1, q < 2p. THEOREMB. There exists a manifold Mm such that I'p+l,q= I(Mm) for p<q, p+q=m. THEOREMC. K1(7rp(SOq.1), Sq+1) = I(SP XS q+1), for p q,p+q i 4. COROLLARY1. Let S14 S14. Then I(S3XS14) contains S17 which does not belong to 017(a7r). COROLLARY2. Let S10 be the generator of the 3-component of 010 N Z23Z3. Then I(S3XS10) is equal to 013. COROLLARY3. I(Sp X S q)= 0 for p+q >_5 therefore I(S 3 X S14) ~ I(S 3 X S 14) and I(S3 X S10) ~ I(S3XS10). These show that the inertia group is neither PL homeomorphism invariant nor tangential homotopy equivalence invariant and that the conjecture of Novikov is negative. COROLLARY4. If Sq is embeddable in Mp+q with trivial normal bundle, N then 1(M) contains K1 wp(SO2), Sq . (Cf. Theorem of Munkres in [7].) REMARK. Smooth structures on S p X S q are completely classified by com- bining Theorem C and the Novikov's work [10]. N 40 K. KAW AKUBO § 3. A lemma. In this section we assume p < q. Let c1 be the zero cross section of the bundle (B2, p2). If the characteristic map h2 of B2 is contained in Image s where s : 1Cq(SOp)--~?Cq(SOp+1), then we can write B2= B2Q+ 01 where Ol denotes the trivial 1-disk bundle and B2 is Dp bundle over S q+1 with the characteristic map h2 rcp(SOq)such that she= h2. The trivial 1-disk bundle Ol can be written as 01= S q+1x I[-1, 1]. Let c~ be the zero cross section of B~ and c2 the cross section of B2 which is written as cl Q 1 using the above 2 expression. In B2, we take two tubular neighborhoods T1 and T2 of c1 and c2 respec- tively such that T1 T2= ~5, T1 C Int B2, T2C Int B2 and T 1n D+ 1 X Dp~-1 = D+ 1X UE, T2 D+ 1x Dp~1= Dq+1 x U e where D+ 1X Dp~1 denotes the first part of B2 and UE and U E are E neighborhoods of 0 X 0 and Ox---1 in Dr' x I [-1, 1] 2 = D1 respectively. Since c2 is diffeotopic to c1, T 1 and T2 are cliffeomorphic to B2.
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