  Based on the algebraic cobordism theory of Levine and Morel, we develop a
theory of algebraic cobordism modulo algebraic equivalence.
  We prove that this theory can reproduce Chow groups modulo algebraic
equivalence and the semi-topological $K_0$-groups. We also show that with
finite coefficients, this theory agrees with the algebraic cobordism theory.
  We compute our cobordism theory for some low dimensional varieties. The
results on infinite generation of some Griffiths groups by Clemens and on
smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in
terms of this cobordism theory.
