On numerical equivalence for algebraic cobordism

We define and study the notion of numerical equivalence on algebraic cobordism cycles. We prove that algebraic cobordism modulo numerical equivalence is a finitely generated module over the Lazard ring, and it reproduces the Chow group modulo numerical equivalence. We show this theory defines an oriented Borel-Moore homology theory on schemes and oriented cohomology theory on smooth varieties. We compare it with homological equivalence and smash-equivalence for cobordism cycles. For the former, we show that homological equivalence on algebraic cobordism is strictly finer than numerical equivalence, answering negatively the integral cobordism analogue of the standard conjecture $(D)$. For the latter, using Kimura finiteness on cobordism motives, we partially resolve the cobordism analogue of a conjecture by Voevodsky on rational smash-equivalence and numerical equivalence.Comment: v1: 14 pages/ v2: 14 pages with obvious silly errors from v1 corrected/ v3: 30 pages. This version is not the final revised and accepted version due to copyright issues. The final accepted version reflecting the referee comments is to appear in J. Pure Appl. Algebr

Algebraic cobordism theory attached to algebraic equivalence

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.Comment: 30 pages. A version of this article was accepted to appear in J. K-theor

Semi-topologization in motivic homotopy theory and applications

We study the semi-topologization functor of Friedlander-Walker from the perspective of motivic homotopy theory. We construct a triangulated endo-functor on the stable motivic homotopy category \mathcal{SH}(\C), which we call \emph{homotopy semi-topologization}. As applications, we discuss the representability of several semi-topological cohomology theories in \mathcal{SH}(\C), a construction of a semi-topological analogue of algebraic cobordism, and a construction of Atiyah-Hirzebruch type spectral sequences for this theory.Comment: v1: 41 pages; v2: 39 pages. The 'idempotence' part of v1 deleted, with some minor revision; v3: 24 pages. Largely rewritten and compactified. A variation of this version is accepted to appear in Algebraic & Geometric Topolog