On the relation of connective K-theory to homology

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Abstract
Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectrainducing a morphismof homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequencewhere Z is viewed as a Z[t] module via the augmentationand, is induced by η*in the natural way.

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