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wise. As an example of such invariants, homdimBPifBP* (X) has been studied by many authors. Especially, Johnson-Wilson [3] proved that homdim^BPiC-JO^n iff BP »(.X)c±BP ,(8)BP,(X) where BP# *( — ) is the bordism theory with the coefficient BP *— BP* Moreover, Johnson-Wilson [4] defined another invariant t(X) as follows: t(X)^n iff there is a jBP*-module isomorphism P(n)*(X) ~P(n)*®H*(X*, Zp) where P (»*(-) is the bordism theory with the coefficient P(n) *~BP*/In = BP*/(p, vl9 ..., i^). This invariant ^(X) appears to have better properties and to be more easily computable than homdim^p^BP* (X). In this paper, we shall study it in comparison with homdimBpj(BPJ|c(X). In § 1 we shall give the definition of t(X) and consider its geometric meaning. In § 2 we shall study the properties of £(X) in connection with skeletons of X, the Spanier-Whitehead duality, cohomolgy operations of H*(X;Zp), the BP^-module structure of X;Zp), the smash product, and the cofibering. I would like to take this opportunity to thank Professor S. Sasao

On an invariant defined by using {$P(n)\sb\ast (-)$} theory