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A particularly nice canonical subspace of      H ∗  ( B O ) {H^{\ast }}(BO)       is defined. The bordism class of a map      f : X →  Y f:X \to Y      , where      X X       and      Y Y       are compact, closed manifolds, can be determined by the characteristic numbers corresponding to elements of this subspace, and these numbers can be easily calculated. As an application, we study the "fixed-point manifold" of a parameter family of self-maps      F : M ×  X →  X F:M \times X \to X      , thus refining to bordism the usual homological analysis of the diagonal which is the basis of the standard Lefschetz fixed point theorem.

A canonical subspace of 𝐻*(𝐵𝑂) and its application to bordism