Vol. 38, No. 2, 1971

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A note on annihilator ideals of complex bordism classes

Larry Smith

Vol. 38 (1971), No. 2, 551–558

Recent studies of the complex bordism homology theory ΩU() have shown that for a finite complex X the integer hom-dim 0UΩU(X) provides a useful numerical invariant measuring certain types of complexity in X. Associated to an element α ΩU(X) one has the annihilator ideal A(α) ΩU. Numerous relations between A(α) and hom-dim l2UΩU(X) are known. In attempting to deal with these invariants it is of course useful to study special cases, and families of special cases. In this note we study the annihilator ideal of the canonical element σ ∈∼NU9Ω(X) where X is a complex of the form

S2N   e2N+1 ∪e2N+2n1−1 ∪ ⋅⋅⋅∪e2N+2nk− 1

and N >> n1,nk > 1, and p an odd prime. We show that A(σ) a [V 2p22 ],,[V 2pS2 ],, where [V 2pS2 ] Ω2pS2U is a Milnor manifold for the prime p. This provides another piece of evidence that for such a complex X, hom-dim ΩUΩU(X) islor2.

Mathematical Subject Classification
Primary: 57C20
Secondary: 57D99
Received: 5 February 1971
Published: 1 August 1971
Larry Smith
Georg-August University
D Gottingen