A Note on the bP-Component of (4n - 1)-Dimensional Homotopy Spheres
Author(s): Stephan Stolz
Source: Proceedings of the American Mathematical Society, Vol. 99, No. 3 (Mar., 1987), pp. 581
-584
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2046369
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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 99, Number 3, March 1987
OF (4n
-
A NOTE ON THE bP-COMPONENT
1)-DIMENSIONAL HOMOTOPY SPHERES
STEPHAN STOLZ
ABSTRACT.
The bP-component of a (4n1 - 1)-dimensional homotopy sphere L: E
04 - I =_bP4,, ?D(CokerJ)4n -1 bounding a spin manifold M is shown to be computable in terms of the signature and the decomposable Pontrjagin numbers of M.
Let 01,- be the group of h-cobordism classes of (m - 1)-dimensional homotopy
spheres and let bP,, c 07,_ be the subgroup of those homotopy spheres bounding
parallelizable m-manifolds. Using results of Kervaire and Milnor [5], G. Brumfiel
showed that 04n-1 has a direct sum decomposition
04ti- I =_bp4n ED7T4til-II/iM(M)
(SO)
where J: -7TZ
fTsis the stable J-homomorphism [1]. The group bP4n is
cyclic and its order IbP4nlcan be expressed in terms of the nth Bernoulli number
(see below). To define the projection map
-
s: 04n -I
bP4n
Z/IbP4nIZ
Brumfiel shows that every homotopy sphere E E 04n-I bounds a spin manifold M
with vanishing decomposable Pontrjagin numbers and that the signature of such an
M is divisible by eight. Then he defines s by s(E) := sign(M) E Z/IbP4jIZ [1].
The above definition is not suitable to compute s(E) for a homotopy sphere E
given explicitly by some geometric construction. The reason is that it is usually not
possible to find an explicit spin manifold bounding E whose decomposable
Pontrjagin numbers vanish. For example, if E is constructed by plumbing it bounds
a manifold M by construction, but in general the decomposable Pontrjagin numbers
of M do not vanish.
In this note we show how to compute s(E) from the signature and the decomposable Pontrjagin numbers of a spin manifold M bounding E. To describe explicitly
which linear combination of decomposable Pontrjagin numbers is involved, let
L(M) (resp. A(M)) be the L-class (resp. the A-class) of M, which are power series
in the Pontrjagin classes of M [4]. For any power series K(M) in the Pontrjagin
classes, let Kn(M) be its 4n-dimensional component. Let ph(M) be the Pontrjagin
character of M, i.e. the Chern character of the complexified tangent bundle of M.
Here we think of the tangent bundle as an element of KO(M), in particular,
Received by the editors December 20, 1984 and, in revised form, January 7, 1986.
1980 MathemnaticsSubject Classification (1985 Revision). Primary 57R60, 55R50.
`(41987American Mathematical Society
0002-9939/87 $1.00 + $.25 per page
581
STEPHAN STOLZ
582
pho(M) = 0. Then define
Sn(M) = 8Ln(M) + bP4a1(cnAn(M)
n(-l)ndn(4(M)ph(M)),),
where an = 1 for n even, an = 2 for n odd, and cw, dn are integers such that
cnnum( Bn/4n) + dndenom( Bn/4n) = 1.
Here BJnis the nth Bernoulli number and num(Bn/4n) (resp. denom(Bn/4n)) denote
the numerator (resp. denominator) of the irreducible fraction expressing the rational
number Bn/4n. We will show below that Sn(M) is a polynomial in P,... I Pn-1'i.e.
Sn(M) does not involve Pn. Note that for 1 < i < n we can interpret p, as an
element of H4i(M, aM), due to the isomorphism H4,(M)_ H4'(M, aM). Hence
Sn(M) E H4n(M, 3M) and we can form the Kronecker product (S,j(M), [M, 3M])
with the relative fundamental class of M.
THEOREM. Let
manifold M. Then
be a (4n - 1)-dimensional homotopy sphere bounding a spin
E
sign(M) - (Sn(M), [M, 3M]) mod|bP4nIZ.
This theorem generalizes some results of R. Lampe [6], who computed the
bP-component of (4n - 1)-dimensional homotopy spheres bounding (2n - 1)connected manifolds, and of G. Brumfiel, who obtained a formula for s(E) - s(E'),
s(E)
=
8
where E, E' are homotopy spheres bounding homotopy equivalent manifolds [2,
Proposition 5.1, Corollary 5.8].
The expression 8 sign(M) - (Sn(M), [M, 3M]) can be viewed as a refinement of
the n-invariant of Eells-Kuiper [3]. They use the integrality of (A(W), [W])/an for
closed spin manifolds W4n to prove that their n-invariantis well defined. We will use
the integrality of (A(W), [W])/an and (A(W) ph(W), [W])/an to show that
sign(M) -(Sn(M), [M, M]) E Z/IbP4n IZ
is independent of the choice of M.
My original motivation for this work comes from the study of highly connected
smooth manifolds. There are classification results for highly connected 'almost
closed' manifolds, i.e, manifolds whose boundaries are homotopy spheres [9, 10]. To
obtain results on closed manifolds, one has to determine whether the boundary of a
given highly connected, almost closed manifold M is diffeomorphic to the standard
sphere. In [8] it is shown that the cokernel J-component of aM often vanishes. Thus
it remains to compute the bP-component which is easily done using the above
theorem if M is 4n-dimensional and using [8, 13] if M is (4n + 2)-dimensional.
PROOF OF THE THEOREM.First we show that Sn(M) does not involve Pn.
An ( M)
phn(M)
('(M)
ph(M))n
=
=-
=
2(2n)!
(2
-
Pn + decomposables,
l)!pn + decomposables,
A'(M)Oph(M)n + '(M)n ph(M)o + decomposables
(2n -1)! Pn+ decomposables.
THE hP-COMPONENT OF HOMOTOPY SPHERES
583
Thus
+ (-I)
c,nAn (M)
(2n1
"dn (A(M)
1)!
(4n
n
ph(M))
+n
(2n - 1)! denom(Bn/4n)
--
(2n
4c,1num 4n) + d4ndenom( ' )
mod decomposables.
~~~~~p
1)! denom(Bn/4n) pn
-
According to [5]
2 2n-2(22n1
1bP4
1) num(4Bn/n).
-
Using the facts that
4I denom( Bn/n)
for n even,
and
2 1denom( Bn/n),
4
+ denom( Bn/n)
for n odd
[7, p. 284] we conclude that
num(4Bn/n) =
an
num(Bn/n) = an num(Bn/4n).
It follows that
(Cn 4n (M)
-
+ (-1I)dn(4(M)
ph(Md)n))
22n-2 (2 2n-1 1 n-2(22n
- )!
1) Bt
41
p, + decomposables.
On the other hand
1
-
L(M)
2-1)
1 22n(2~"
8
(2)
=
1) B,1+ decomposables
[4, p. 12], which shows that S,1(M) is a polynomial of P1,P'. .
The next step is to prove the equality
S(s)
= ' sign(M) -(Sn(M),
[V, aM])
mod|bP4j iZ.
If the decomposable Pontrjaginnumbers of M and hence (S,1(M), [M, aM]) vanish,
the above equation holds by the definition of s. Thus we have to show that the
right-hand side is independent of the spin manifold M. Let N be another spin
manifold bounding E and let W be the closed spin manifold obtained by gluing M
and -N along E. Then
sign(W)
= sign(M)
-
sign(N)
and
(S, (W), [W])
=
(S, (M), [M, M]) -(S,(N),
[N, N])
584
STEPHAN STOLZ
It follows that
I
sign(M)
-
-
(Sn(M), [M, aM])
sign(N)
-
-
Sn (N), [N, aN]))
sign(W) - (Sn(W), [WI)
8(sign(w)
-
(Ln(W),
1
+jbP4nIcCn
[WI))
A
A n(W),
[W])
+(-1)
ndna
A~~~~~~~
(A(W)ph(W))n,
[W]))
=
mod IbP4nIZ since sign( W) = ( Ln(W), [W]) by Hirzebruch's signature theorem and
since (An(W),[W])/an
resp. ((A(W)ph(W))n, [W])/a,, are integers by the
Hirzebruch-Riemann-Roch Theorem [4, Theorems 26.3.1 and 26.3.2]. Q.E.D.
REFERENCES
1. G. Brumfiel, On the homotopygroupsof BPL and PL/O, Ann. of Math. (2) 88 (1968), 291-311.
, Homotopy equivalences of almost smooth manifolds, Comment Math. Helv. 46 (1971),
2.
381-407.
3. J. Eells and N. Kuiper, An invariantfor certainsmooth manifolds, Ann. Mat. Pura Appl. 60 (1962),
93-110.
4. F. Hirzebruch, Topologicalmethods in algebraic geometry, 2nd corrected printing of the 3rd ed.,
Springer, 1978.
5. M. Kervaire and J. Milnor, Groupsof homotopyspheres. I, Ann. of Math. (2) 77 (1963), 504-537.
6. R. Lampe, Diffeomorphismenauf Spharen und die Milnor-Paarung,Diplomarbeit, Mainz, 1981.
7. J. Milnor and J. Stashoff, Characteristicclasses, Ann. of Math. Studies, no. 76, Princeton Univ.
Press, Princeton, N. J., 1974.
8. S. Stolz, HochzusammenhdngendeMannigfaltigkeitenund ihre Rdnder, Lecture Notes in Math., vol.
1116, Springer-Verlag,Berlin and New York, 1985.
9. C. T. C. Wall, Classification of (n - 1)-connected 2n-manifolds, Ann. of Math. (2) 75 (1962),
163-189.
10.
, Classificationof (n - 1)-connected(2n + 1)-manifolds, Topology 6 (1967), 273-296.
FACHBEREICH MATHEMATIK, UNIVERSITAT MAINZ, SAARSTR. 21, D - 6500 MAINZ, WEST GERMANY
Currentaddress: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556