Complex bordism

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1 Introduction

Complex bordisms (also known as unitary bordisms) is the bordism theory of stably complex manifolds. It is one of the most important theory of bordisms with additional structure, or B-bordisms.

The theory of complex bordisms is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordisms or other bordisms with additional structure (B-bordisms). Thanks to this, the complex cobordism theory found the most stricking and important applications in algebraic topology and beyond. Many of these applications, including the formal group techniques and Adams-Novikov spectral were outlined in the pioneering work [Novikov1968].

2 Stably complex structures

A direct attempt to define the bordism relation on complex manifolds fails because the manifold $W$ is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of the complex structures. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.

9 References

• [Novikov1968] Template:Novikov1968

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