Torsion in the B-bordism group for connective covers of BO

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[edit] 1 Question

For each connective cover of BO, i.e. (n-1)-connected space BO\langle n \rangle with a fibration BO\langle n \rangle \to BO inducing an isomorphism on homotopy groups \pi_k for k\ge n, there is a normal bordism group \Omega_*^{BO\langle n \rangle} as defined in the article on B-bordism. (The first examples are unoriented bordism, oriented bordism, spin bordism and string bordism.)

Question 1.1. Which primes appear in the torsion part of \Omega_*^{BO\langle n \rangle}?

[edit] 2 Answers

There is 2-torsion in \Omega_*^{BO\langle n \rangle} for all n. For p odd there is p-torsion in \Omega_*^{BO\langle n \rangle} iff 2p-2 < n.

[edit] 3 Proof

Theorem 3.1 [Serre1951]. There is p-torsion in the framed bordism group (stable homotopy group of spheres) \Omega_{2p-3}^{fr}=\pi_{2p-3}^{st}.

Corollary 3.2. For 2p - 2 < n there is p-torsion in \Omega_*^{BO\langle n \rangle}.

For the proof one shows that for k\le n-2, \Omega_{k}^{BO\langle n \rangle}\cong \Omega_{k}^{fr} which follows from obstruction theory. Thus there is p-torsion in \Omega_{2p-3}^{BO\langle n \rangle} for 2p-1 \le n.

We denote by A_p the mod p Steenrod algebra and by A'_p the subalgebra of the mod p Steenrod algebra generated by reduced powers. It is isomorphic to the quotient algebra A_p/ \langle \beta \rangle of the Steenrod algebra by the two-sided ideal generated by the Bockstein operation.

Theorem 3.3 [Milnor1960]. Let Y be a connective spectrum with finitely many cells in each dimension. If H^*(Y;\Zz_p) is a free A_p/ \langle \beta \rangle-module with even-dimensional generators, then \pi_*Y contains no p-torsion.

This is proved by finding a rather explicit free resolution of H^*(Y;\Zz_p) over the Steenrod algebra, and showing that the Adams spectral sequence for [M(p),Y] degenerates at the E_2-term, where M(p) is the mod p Moore spectrum.

Theorem 3.4 [Giambalvo1969, Corollary 1]. For p>2k, the module H^*(MO\langle 4k\rangle;\Zz_p) is a free module over A'_p generated by the Thom class.

This uses the computation of H^*(BO\langle 4k\rangle;\Zz_p) as a Hopf algebra, the work by Milnor-Moore on Hopf algebras and the action of the Steenrod algebra on Pontryagin classes.

Corollary 3.5. For p>2 and 2p-2 \ge n there is no p-torsion in \Omega_*^{BO\langle n \rangle}.

Here we use in addition to the two previous theorems that for p>2 the p-torsion in \Omega_*^{BO\langle 4k \rangle} agrees with the p-torsion in \Omega_*^{BO\langle 4k-i \rangle}, i=1,2,3 since the higher connective covers are obtained by possibly taking homotopy fibers of maps to K(\Zz_2,j)'s.

[edit] 4 Further discussion

It seems much harder to say more about the possible orders of torsion elements in \Omega_*^{BO\langle n \rangle}. It is known that in unoriented, oriented and spin bordism each torsion element has order 2, and that in string bordism the order of each torsion element divides 24.

[edit] 5 References

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