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 topologically) 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 eomorphism 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 equivalent (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 diffeomorphism 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 homotopy between h and h'. We write H(x, t) as hr(x). Let r' E I'q+1 be other representative 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_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 respectively 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. Let X = B1 V B2. We connect dT1 and aX by an imbedding 1: 1 [0, 1] -- X such that l(IntI)nT1=~5, 1(1)nT2=~b, l(IntI)naX=~5, 1(1)nT1=l(0), 1(1) FIX= 1(1) and l(I) is contained in (Int Dq~')X Dpi 1. We take a tubular neighborhood T of l(I) in X-Int T1-T2, which is clearly cliffeomorphic to I x D~'. Let Tl = T1U T. Let Y denoteX- (IntT ; U 1(1)x Int DP~1~1). LEMMA. Y is diffeomorphic to T2 for dim Y ? 6. PROF. In case where p > 1. According to Smale [10], if the natural inclusion c : T2 -~ Y induces a homotopy equivalence and rc1(Y-1nt T2) = rc1(FT2) =rr1(dY)= {1}, then TZ is diffeomorphic to Y. It is easy to see that 2cfY--IntT2) = ir1(FT2) = 2r1(FY) = {1}. Firstly we prove that c : T 2-~ Y induces an isomor- phismof homologygroups. We put Y'= FIT-(1(1)x Int D). We shall examine the next commutative diagram.

---> H~~1(Y, Y') - - Hti(Y') _- H1(Y) _- Hz(Y, Y') _-

1 N N (*) --~ Hti+1(X, T ~) --~ Hf T D -> HE( X) ---.-> HfX, T 1) --~ where HfY, Y') is isomorphic to H1(X, T 1) by the excision isomorphism. 1 < i < q : It is easy to see that Hf Y) N 0 from the diagram (*). i = q+1: In the diagram (*), putting i = q+1, the isomorphisms Hq+1(Y') N Hq ~1(Ti) M Z and Hq(Y') N Hq(T Q N 0 hold. Hence we obtain the isomorphisms Hq+1(Y) N Hq+1(X) N Z by the five lemma. The composition c' o c of the natural The inertia groups of homology tori 41

maps Hq+1(T2)-`*Hq+1(Y)--+Hq~.1(X) is an isomorphism and c' is an isomorphism from the above. Consequently c is an isomorphism. q+2 < i : The isomorphisms HA(Y) H(X) 0 are easily deduced from the diagram (*) likewise. On the other hand H2(T2) is zero except for i= 0, q+1. Thus we can conclude that the natural inclusion c : T2 --~ Y induces the isomorphism of homology groups. Hence the natural inclusion c11: aT2 -> Y-Int T2 induces the isomorphism of homology groups by the excision isomorphism i. e. c" gives a homotopy equivalence. Making use of the Poincare- Lef schetz duality theorem, the natural inclusion Cm: a Y -> Y-Int T2 induces the isomorphism of homology groups i. e. c"' gives a homotopy equivalence (see J. H. C. Whitehead [19]). Hence (Y-Int T2, aT2, aY) is an h-cobordism and T2 is dill eomorphic to Y. In case where p=1: Since 2q(SO2) = 0 for q >_ 2, one has T 2 = S q~-1X D2. Since 7r1(X) = {1} and Y= X- (IntT ; U1(1)x Int Dq~2), any elementof 7r1(Y)is homotopicinto aT -l(1)xIntDq~2 by the Van Kampen's theorem. As the generator of 7r1(aT1-l(1) XInt Dq+21 ^'Tr1(aTi-Int D+2) 7r1(Sq+1X S 1-Int Dq ~2) ^'Z, onecan 1 take a fibre * x S 1 of aT i. But this is homotopic to aD2 X 0 when we write B1 as D2 X Dq~1 U D2 X Dq~1 and homotopic to zero in Y. On the other hand, one has 7r1(3T2)= 7r1(Sq+1 X S 1) . Z and 7r1(aY) = 7r1(aTi aX) N Z. Since the generator of 7r1(dY) is homotopic to the generator of ?r1(dT2) in (Y--Int T2) and 7T1(Y)= {1} we obtain 7r1(Y-Int T2) ^' Z. Apparently inclusions of universal coverings aT2 -~ Y-Int T 2 and a Y --~ Y-Int T 2 induce homology isomorphisms. Thus inclusions aT2-- Y-Int T2 and dV--j Y-Int T2 are homotopy equivalences (see J. H. C. Whitehead [19]). When 'r1 "' Z, Whitehead group is trivial and s-cobordism theorem (M. Kervaire [4]) implies that V--Int T2 = aT2 X I. Con- sequentely Y is dilleomorphic to T2. This completes the proof of Lemma.

§ 4. Proof of Theorems.

(a) PROOF OF THEOREM A. Firstly we shall prove this theorem when Mm boundsa 7r-manifoldWm1 1which is [_m±1 -connected.If S E I(M), there 2 exists a dill eomorphism f : Mm-Int Dm -~ Mm--Int Dm such that f dDm 1, represents S (see I. Tamura [13]). (Here we identified Fm and em by the theorem of Smale.) Using this dill eomorphism f, we construct a manifold W U W which is denoted by X. Clearly the boundary aX is cliffeomorphic to r

N S. One has easily 7r1(X) = {1} by the Van Kampen's theorem. Making use of the Mayer-Vietoris exact sequence --~H1(M-Int D) ->H2(W) 0+ H1(W) ->H(X) --~ 42 K. KAWAKUBO

Hti_1(M--Int D) - and the Poincare-Lefschetz duality theorem, Hi(W) Hm+1-i(W,M), it is easy to see that z i=0 H(X)N Z0+ ... ~Z i=p+1, q 0 otherwise. Let a1',• •. , ak Hp(M) and bi, •• • , bk Hq(M) (for some k) be bases whose inter- section numbers are ai o b~= ai;. Let f i : S p -> M, i =1, •. • , k be the mapping such that [ f i(Sp)] = a' where [ f i(Sp)] denotes the homology class represented by f(S). By Whitney's imbedding Theorem [20], we may suppose that f i (i =1, ..•, k) are imbeddings and f (S) n f ; (Sp) = ~5 i *j. Let i : M--Int D -j W be a natural inclusion map. Since i*[ f i(Sp)] = 0 and i* f *[f '(Sp)] =0, we can extend f i and f o f i to ft: Dp+1--~W and f i : Dp+1- W. These give an imbedding f1: 5p+1-~ X such that [ f i(Sp+1)]= a1, where ai is a generator of Hp 1(X) such that aati = ai where a is a boundary homomorphism of Mayer-Vietoris exact sequence :

-~H~,.1(W) +OHp1(W)-- H~+1(X)--pHp(M--Int D)-~HA(W)3Hp(W)-~ Zt 11 11 1Z 0 0 0 0 Here we may assume that Sp+1 is a natural sphere and f 1(S) n f (S1) = (i i ~ j. Let N(f1) be a tubular neighborhood of f 1(S') in Int X (i =1, ..•, k) such that N(f 1)r N(f) = ~5 i =j. N(f1) is a Dq-bundle over Sp+1; (N( f2), sp+1, Dq, pi). Let X = N(f1)Ii ... b N(f) C Int X be a boundary connected sum of N(f 1), ..•, N(f) in Int X. According to Smale [12], we have a handlebody decomposition as follows : X= (N(f1)• .. N(fk)) U Dqx Dp+1U...U Dk x D+~,

and that we can suppose that the handle Dq x Dp+1 represents the homology class bi (i =1, ..•, k) where b1 denotes the image of b' by the natural isomorphism Hq(M) --~ Hq(X ). Thus the homotopy type of X is given by XSN +1V V S p+1U eq U ... U e. Since each e4 attaches to equators of S p+' V •• • V S k+', X has the homotopy type X N S f1 V••• V S r' V S g VV S k. According to I. Tamura[15], X canbe writtenas (iv(f1)N(g1) b ... (N(fk)VN(gk)). (Where gi is an imbedding of the homology generator b..) Since we can write a N(f1)VN(g1)) =K2(hi, h2)where h; ~p(SOq),hZ2rq _1(SOp+1)arecharacteristic maps of the bundles N(f1) and N(g1) respectively, I(Mm) Thus Theo-

remA is provedwhen Mm bounds a 7r-manifold Wm+1 which is m[_Ti]con- -

2N The inertia groups of homologytori 43

netted. Secondly we shall prove that the general case is reduced to the case above. One has easily that I(M) = I(M S ) and I(M) +I(M') C I(M M'), therefore if one proves that 1VI S or M M S bounds a 2r-manifold W which is [m]- ]1 -connected,theproof of TheoremA is complete.If m 8k+6,there 2 exists a homotopy sphere S such that M S is a boundary of a 7r-manifold W. (Cf. E. H. Brown and F. P. Peterson [2].) N N If m = 8k+6 there exists a homotopy sphere such that M S or M M S is a boundary of a 7r-manifold W. At first we will kill the fundamental group of W. H1(W m+1) for i<_Cm±1- ]-i canbe killed by surgeriesinductively. Case m+ 1= 2n+1. Since Hn(a W) N Hn,.1(a W) 0, H(W) can be killed (see C. T. C. Wall [17]). Case m]-1= 4n, there exists a 7r-manif old W' such that index W' = -index W and d W' is a homotopy sphere and that HfW') = 0 for 1 <_ i _<2n-i. Since H2na(W b W'))N H2n-i a(W b W')) 0 andindex W W'=0, we can kill H2n(W b W') completely by surgeries (see J. Milnor [9]). Caser+1= 4n+2.H2n+1(W IiW) canbe killed,since H2n+1(a(W b W)), =H2-1(M M) ^' 0, H2n a(W i W)) =H2n(M M) 0 andArf invariant ofT4T J W is zero. Consequently M S or M M S for some S m bounds a 7r-manifold Wm+1 which is Cm--1- ]-connected. Thus Theorem A is completelyproved. (b) PROOF OF THEOREM B. Let ati = K2(hi, she) E Tp+1,qfor h 7rp(SOq) h2 7rq_1(SOp)i=1, ••• , k such that {a1 (i=1, ••• , k)} generate Fp+a,q. Let (B1, Sp+1,Dq, pi) and (Bi, S q, Dp 1, pi) be disk bundles over spheres with characteristic maps h;, she respectively. We denote B1 Bi by Xi. By Lemma in § 3 Xti can be written as T1 U Ti where T1 is a disk bundle over sphere with Fz characteristic map she and Fi is an orientation reversing diff eomorphism Fi : aT1-Int D -> Ti-Int D. Since there is an orientation reversing diff eomorphism R : T1->-T1 using a cross section, a1= aX1= a(T1 U T1) = a(T1 U --T2). Fi F1R Hence I(13T1) contains ai. Thus if we take ST,, •• • aTk as M, I(M) = I(aT, aTk) DI(6T1)+ +I(aTk) = T'p+l,q. Next we shall prove that reversed inclusion I(M) C F +,,q holds. For any element a E I(M), there is a dilleomorphism F : M- Int D -->M- Int D, such that F aD F p+q represents a. We put T = T 1 b Tk. Then we construct a manifold X such that X = T U T where F is the

F abovemap. SinceHfT) is clearlyzero for i <_ m+[L],x= TU T can be 2 h written as X = (B1 U T1) b ... b (Bk v T k) by the analogous method in the proof

2 44 K. KAWAKUBO

of Theorem A, where Bi is a disk bundle over sphere with some characteristic map hit E 7cp(SOq) i -=1, •. • , k. Hence a = aX is contained in Thus

I(M)=r,.-' p+lq• N (c) PROOF OF THEOREMC. When p > q, we have Sp X Sq+1= Sp X S q+l. (See W. C. Hsiang and J. Levine and R. H. Szczarba [3].) From the proof of Corollary 3 in the later, we have I(SP xS q+1)= I(SP x Sq+l) = {0}. On the other handKl (2cp(SOq+I), Sq+l is containedin I(SP xS q+l) by Lemma.Therefore Theorem C trivially holds when p >_ q. Now we may assume p < q. First we shallprove that I(Sp x S q+l) contains Kl (icp(SOq~i), Sq+l Leta= K1(h,Sq+l) Kl(7 p(SOq+I), Sq+1). We now construct two disk bundles, B1, B2 as follows. Let (B1, Sp+1,D'', p1) be a disk bundle with a characteristic map h, and (B2, S q+l, Dp+l, p2) be the trivial bundle over a homotopy sphere S'1'. On the other hand the pairing Kl can be interpreted as follows. Let r ~q+1 be a corre- sponding element of S q+' Oq+l, One defines the cliffeomorphism F': Sp x S'1 --jSp X S q byF'(x , y)= (x,rh(x)y . Attachingtwo manifolds W1=Dp+l XS q and W2 = S p x D q+1 by the cliff eomorphism F': SpxSq -~ S p x S q, we have a homotopy sphere .' - = Dp+l X S q U S p X Dq+1 for p < q. Then ,'' is cliff eomorphic to F' ' by the cliffeomorphism f : Dp+' X S q U SpXDq+l --~Dp+1 x S q U Sp X Dq+1 defined F F' by f= id x r-1 on Dp+l x S q and f = id X id on SpXDq+l . It is easy to see that f is a diff eomorphism between ~' and ~?'. Let G1 be a dilleomorphismG1:Sp x Dq+i-; Sp x Dq+' defined by G1(x,y)= (x,h(x)y). Let B1 = D++1 x D'~1 U Dp+1 X Dq+l. Let G2 be a dif eomorphism G2: S q x Dp+l G1 -~S q x Dp+ldefined by G2(x,y) = (r(x),y). Let B2= Dq+1x Dp+l U DP1x G2 We define B1 °, B2 to be the oriented differentiable (p+q+2)-manifold obtained as a quotient space of B1 U B2 by identifying Dp+l x Dq+l of B1 and D+ t X Dp+l of B2 in such a way that (x, y) = (y, x) (x E Dp+l = Dp+l, y E = D1). Let G1= G1 Sp X Sq and G2 = G2 Sq x Sp. Using a dif eomorphism R : SpxSq -~ Sq x Sp defined by R(x, y) = (y, x), we define G2 = R-1G2R. Then we have F' = G2 Gi and (B19. B2) = Dp+i x S q U Dq+t x S p = Dp+l x S q U Dq -1x Sp = ~' = ~'. Thus a G2G1 F , can be written as a(B1 B2). Considering a trivial Sp bundle over Sq+l in place of Sp bundle over Sq+1 of Lemma, quite analogously, one has that B1.9- B2 is cliffeomorphic to B2 U B2 where H is a dilleomorphism H : aB2-IntD -> B2--IntD. jr

„ The inertia groups of homology tori 45

Therefore a = a(B1 B2) = a(B2 U B2) implies that the inertia group of aB2 =

H S p X S q~1 contains a. Conversely for any element a E I(S p XS q+l), there is a dill eomorphism H: S PX S q+1_ Int D -- S PX S 2+l- Int D such that H ~aD Tp+q+l

N represents a. Using this dill eomorphism we construct a manifold D'x'

N U Dp~l X S q~1 which is denoted by X. Clearly aX = a and like the proof of

H Theorem A, we can prove that X can be written as B1 (S' 1X Dp+1) where B1 is a disk bundle over sphere with some characteristic map h 7rp(SOq+I). This implies a = K1(h,S q+1) and completes the proof of Theorem C.

§ 5. Proof of Corollaries. (a) PROOF OF COROLLARY1. Since the pairing K1 is equal to that of Novikov (see [111 p. 235), next diagram commutes up to sign. K1 lTp(SOq+l)X rq+l -~ rp+q+l JpX~ w Gp X Gq+1/Im J q+1 ' Gp+q+ilImJp+q;-1 C where Gi is the stable homotopy group G1= 7r1+k(Sk) and w is the KErvaire- Milnor map [5] and Jp denotes the Hopf-Whitehead homomorphism and C is the composition. The Kervaire-Milnor braid

shows that w(S14) (where S 14 S 14) is , or +62 (in Toda's notation) since q5(U2)~ 0 (see Levine [8]). But according to Toda's tables [16], y o (A+62) 46 K. KAWAKUBO = vo i 0 (modIm J). HenceI(S 3 X S 14) = K1 (2r3(SO), S14) is notcontained in 017(a2r). This makes the proof complete. (b) PROOF OF COROLLARY2. Analogously, for the generator S 1° of the

N threecomponent of 010= ZZ+E Z3, K1(7r3(SO), S1° = 013(cf. S. P. Novikov[11] and A. Kosinski [6]). So by Theorem C, we have that I(S 3x S 10) is equal to the whole group 013. (c) PROOFOF COROLLARY3. Let S p 53E I(Sp X S q). Then there is a diff eomorphism f : Sp XSq-Int D -->Sp X Sq-Int D such that f JaD Fp.,q represents Sp+q. We now construct a manifold Dp+1x S q U Sp x Dq "1 which is denoted by f X. If p < q, the homology groups of X are zero except for dimension zero. On the other hand ~r1(X) = ~c1(Sp~q)= {1}. Hence X is dill eomorphic to a disk and Sp+q= aX is a natural sphere. This implies that I(Sp XS3) = 0 for p _6. Therefore I(S 3x S 14)* I(S 3 x S 14)and I(S 3 x S 10) I(S 3x S 10) These show that the inertia group is neither PL homeomorphism invariant nor tangential homotopy equivalence invariant. Since I(S 3 XS 14) is not contained in 017(a~r),the conjecture of Novikov is negative. (d) PROOF OF COROLLARY4. This is obtained as an easy application of Theorem C. Osaka University

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