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

## Contents |

## [edit] 1 Question

For each connective cover of , i.e. -connected space with a fibration inducing an isomorphism on homotopy groups for , there is a normal bordism group 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 ?

## [edit] 2 Answers

There is -torsion in for all . For odd there is -torsion in iff .

## [edit] 3 Proof

**Theorem 3.1** [Serre1951]**.**
There is -torsion in the framed bordism group (stable homotopy group of spheres)
.

**Corollary 3.2.**
For there is -torsion in .

For the proof one shows that for , which follows from obstruction theory. Thus there is -torsion in for .

We denote by the mod Steenrod algebra and by the subalgebra of the mod Steenrod algebra generated by reduced powers. It is isomorphic to the quotient algebra of the Steenrod algebra by the two-sided ideal generated by the Bockstein operation.

**Theorem 3.3** [Milnor1960]**.**
Let be a connective spectrum with finitely many cells in each dimension.
If is a free -module with even-dimensional generators,
then contains no -torsion.

This is proved by finding a rather explicit free resolution of over the Steenrod algebra, and showing that the Adams spectral sequence for degenerates at the -term, where is the mod Moore spectrum.

**Theorem 3.4** [Giambalvo1969, Corollary 1]**.**
For , the module is a free module over
generated by the Thom class.

This uses the computation of 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 and there is no -torsion in .

Here we use in addition to the two previous theorems that for the -torsion in agrees with the -torsion in , since the higher connective covers are obtained by possibly taking homotopy fibers of maps to 's.

## [edit] 4 Further discussion

It seems much harder to say more about the possible orders of torsion elements in . 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

- [Giambalvo1969] V. Giambalvo,
*The cohomology of*, Proc. Amer. Math. Soc.**20**(1969), 593–597. MR0236913 (38 #5206) Zbl 0176.52601 - [Milnor1960] J. Milnor,
*On the cobordism ring and a complex analogue. I*, Amer. J. Math.**82**(1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702 - [Serre1951] J. Serre,
*Homologie singulière des espaces fibrès. Applications*, Ann. of Math. (2)**54**(1951), 425–505. MR0045386 (13,574g) Zbl 0045.26003