Abstract. We define and study the notion of numerical equivalence on algebraic
cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence
of a smooth projective variety is a finitely generated module over the Lazard ring, and
that it reproduced Chow group modulo numerical equivalence.
We compare it with the notions of homological equivalence and smash-equivalence
for cobordism cycles, using Kimura fitniteness on cobordism motives. We resolve part
of the cobordism analogue of a conjecture of Voevodsky on smash-equivalence and
numerical equivalence.