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In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya C1-manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this...
The following is a long-standing open question: "If the zero-framed surgeries on two knots in the 3-sphere are integral homology cobordant, are the knots themselves concordant?" We show that an obvio...
In this paper we introduce the models for $(\infty, n)$-categories which have been developed to date, as well as the comparisons between them that are known and conjectured. We review the role of $(\i...
We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S) there is a unique morphism g : MSp -> BO of commutative ring T- spectra which sends...
We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T^{2}- spectra. The spectrum MSp comes with a natural symplectic orientation given eithe...
We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.
We review the SO(3) instanton gauge theory of Fintushel and Stern and recast it in the context of 4-manifolds with cylindrical ends. Appli-cations to the Z/2 homology cobordism group of Z/2 homology 3...
We show the T2-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the T2-cobordism class of CP2. The main tool is the theory of quasitoric manifolds.

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