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

## 1 Question

For each connective cover of $BO$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}BO$, i.e. $(n-1)$$(n-1)$-connected space $BO\langle n \rangle$$BO\langle n \rangle$ with a fibration $BO\langle n \rangle \to BO$$BO\langle n \rangle \to BO$ inducing an isomorphism on homotopy groups $\pi_k$$\pi_k$ for $k\ge n$$k\ge n$, there is a normal bordism group $\Omega_*^{BO\langle n \rangle}$$\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}$$\Omega_*^{BO\langle n \rangle}$?

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

## 3 Proof

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

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

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

We denote by $A_p$$A_p$ the mod $p$$p$ Steenrod algebra and by $A'_p$$A'_p$ the subalgebra of the mod $p$$p$ Steenrod algebra generated by reduced powers. It is isomorphic to the quotient algebra $A_p/ \langle \beta \rangle$$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$$Y$ be a connective spectrum with finitely many cells in each dimension. If $H^*(Y;\Zz_p)$$H^*(Y;\Zz_p)$ is a free $A_p/ \langle \beta \rangle$$A_p/ \langle \beta \rangle$-module with even-dimensional generators, then $\pi_*Y$$\pi_*Y$ contains no $p$$p$-torsion.

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

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

This uses the computation of $H^*(BO\langle 4k\rangle;\Zz_p)$$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$$p>2$ and $2p-2 \ge n$$2p-2 \ge n$ there is no $p$$p$-torsion in $\Omega_*^{BO\langle n \rangle}$$\Omega_*^{BO\langle n \rangle}$.

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

## 4 Further discussion

It seems much harder to say more about the possible orders of torsion elements in $\Omega_*^{BO\langle n \rangle}$$\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.