Contemporary Mathematics
Volume 19, 1983
ON WEAKLY ALMOST COMPLEX MANIFOLDS WITH
VANISHING DECOMPOSABLE CHERN NUMBERS.
ANDREW BAKER
Abstract:
We describe the subgroup of the complex bordism ring consisting of
elements with only the indecomposable Chern monomial giving a non-
zero Chern number. In dimensions 4k
+
2 we recover results of Ray,
and in dimensions 4k we prove a conjecture of Dyer.
In this note we will investigate the subgroup of the complex bordism
ring MU* consisting of classes for which the only non-zero Chern number is
that coming from the top diMensional Chern class. In dimensions of fom
4k
+
2, we recover results of [Ra]; in dimensions of form 4k, we prove an
old conjecture of E. Dyer [Dy].
Theorem Let Xn
e:
MU 2n be in the subgroup of eleMents for which all
decomposable Chern numbers are zero. Then Xn is a generator if and only
if cn(Xn) takes the value (up to sign)
2, if n
=
1,
(2k)!, if n 2k
+
1, k 1,
dk(2k- 1)!, if n
=
2k, where
~
is the denominator of
B2k/2k, for B 2k the 2k-th Bernoulli number.
1980 Mathematics Subject Classification. 55R50, 57R77
1
©
1983 American Mathematical Society
0271-4132/83 $1.00
+
$.25 per oage
http://dx.doi.org/10.1090/conm/019/711038
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