The composite decomposition and the rational homology cobordism group

Abstract A primary part in the rational homology cobordism group is generated by manifolds for which the order of the first homology is a power of a given prime. Kim and Livingston proved that the rational homology cobordism group does not equal the sum of primary parts. The current paper defines composite parts, generated by manifolds for which the order of the first homology is a product of powers of primes in a given set. We show that the rational homology cobordism group is not generated by composite parts with bounded number of distinct prime factors. Moreover, we give a necessary and sufficient condition for two sums of composite parts to be equal. We also prove the rational homology cobordism group is not right primary decomposable, answering a question of Cha.