Bordism

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1 Introduction

Basic facts from the theory of bordisms and cobordisms are summarised here. For the more specific information, see the pages on unoriented, oriented and complex bordism.

The theory of bordism is is one of the most deep and influential parts of the algebraic topology, which experienced a spectacular development in the 1960s. The main introductory reference is the monograph [Stong1968].

2 The bordism relation

All manifolds here are assumed to be smooth, compact and closed (without boundary), unless otherwise specified. Given two $n$-dimensional manifolds $M_1$ and $M_2$, a bordism between them is an $(n+1)$-dimensional manifold $W$ with boundary, whose boundary is the disjoint union of $M_1$ and $M_2$, that is, $\partial W=M_1\sqcup M_2$. If such $W$ exists, $M_1$ and $M_2$ are called bordant. The bordism relation splits manifolds into equivalence classes (see Fig.~\ref{trcob}), which are called bordism classes.

2.1 References

• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010

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