Fe b 20 03 Torsion primes in loop space homology
نویسندگان
چکیده
We construct a finite 1-connected CW complex X such that H * (ΩX; Z) has p-torsion for the infinitely many primes satisfying p ≡ 5, 7, 17, 19 mod 24, but no p-torsion for the infinitely many primes satisfying p ≡ 13 or 23 mod 24. A homology torsion prime for a topological space Y is a prime p such that H * (Y ; Z) has p torsion. In [3] and [4] respectively, D. Anick and L. Avramov constructed simply connected finite CW complexes X of dimension 4 such that H * (ΩX, Z) has p-torsion for all primes p. Since there are only countably many homotopy types of simply connected finite CW complexes, this immediately raised the Question. If P denotes the set of all primes, what are the conditions for a subset D ⊂ P, to be the set of homology torsion primes of the loop space of a finite simply connected CW complex ? Of course, any finite set can be realized. It suffices to take a product of Moore spaces. On the other hand by a result of McGibbon and Wilkerson [7], if X is rationally elliptic (i.e., dim π * (X) ⊗ Q < ∞), then the set of homology torsion primes for ΩX is finite. In this note we adapt the construction of Anick to prove Theorem 1. There exists a 4-dimensional finite simply connected CW complex X such that the homology torsion primes for ΩX are an infinite set in P, with an infinite complement .
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