# String bordism

## 1 Introduction

$String$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}String$-bordism or $O\!\left< 8 \right>$$O\!\left< 8 \right>$-bordism is a special case of a B-bordism. It comes from the tower of fibrations below.

$\displaystyle \xymatrix{BO\!\left< 8 \right>\ar[d] & \\ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \\ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \\ BO\ar[r]^{w_1}& K({\mathbb Z}/2,1) }$

In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$$BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$$BSpin$ given by half of the first Pontryagin class. The name $String$$String$-group is due to Haynes Miller and will be explained below.

## 2 The String group

There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces. A more geometric way to think about $String(n)$$String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$$K({\mathbb Z},2)$. One may think of $K({\mathbb Z},2)$$K({\mathbb Z},2)$ as the realization of $S^1$$S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$$A_\infty$ space $String(n)$$String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$$Spin(n)$ by the finite dimensional Lie groupoid $S^1$$S^1$ (see [Schommer-Pries2009]). A more explicit model for this extension can be found in [Meinrenken2003].

## 3 The bordism groups

The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]) and the 3 torsion is annihalated by multiplication with 3 (see [Hovey1997]). Moreover, the bordism groups $\Omega_{k}^{String}$$\Omega_{k}^{String}$ are finite for $k=1,2,3$$k=1,2,3$ mod 4.

Clearly, since $BO\!\left< 8 \right>$$BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the framed bordism groups. The first 16 bordism groups have been computed by Giambalvo [Giambalvo1971, p. 538]:

• $\Omega_7^{String} = 0$$\Omega_7^{String} = 0$.
• $\Omega_8^{String} \cong \Zz \oplus \Zz/2$$\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
• $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$$\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by exotic 9-spheres.
• $\Omega_{10}^{String} \cong \Zz/6$$\Omega_{10}^{String} \cong \Zz/6$, generated by an exotic 10-sphere.
• $\Omega_{11}^{String} = 0$$\Omega_{11}^{String} = 0$.
• $\Omega_{12}^{String} \cong \Zz$$\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$$8 \times 992$.
• $\Omega_{13}^{String} \cong \Zz/3$$\Omega_{13}^{String} \cong \Zz/3$, generated by an exotic 13-sphere.
• $\Omega_{14}^{String} \cong \Zz/2$$\Omega_{14}^{String} \cong \Zz/2$, generated by the exotic 14-sphere.
• $\Omega_{15}^{String} \cong \Zz/2$$\Omega_{15}^{String} \cong \Zz/2$, genreated by an exotic 15-sphere.
• $\Omega_{16}^{String} \cong \Zz^2$$\Omega_{16}^{String} \cong \Zz^2$.

At the prime 3 the first 32 bordism groups can be found in [Hovey&Ravenel1995]. Further calculations have been done in [Mahowald&Gorbounov1995].

## 4 Homology calculations

### 4.1 Singular homology

The cohomology ring $H^*(BString,{\mathbb Z}/p)$$H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$$p=2$ by Stong in [Stong1963]:

$\displaystyle H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 4]\otimes H^*(K({\mathbb Z},8))/Sq^2(\iota).$

Here, $\sigma_2$$\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$$\theta_i$ come from the cohomology of $BO$$BO$ and coincide with the Stiefel-Whitney up to decomposables.

From Stong's proof it follows that $H^*(BO)\to H^*(BString)$$H^*(BO)\to H^*(BString)$ is surjective, and $H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 3]$$H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 3]$ is a polynomial algebra.

For odd $p$$p$ the corresponding result has been obtained by Giambalvo [Giambalvo1969].

### 4.2 K(1)-local computations

$K(1)$$K(1)$ locally $MString$$MString$ coincides with $MSpin$$MSpin$ and decomposes into a wedge of copies of $KO$$KO$. However, it is not an algebra over $KO$$KO$. Its multiplicative structure for $p=2$$p=2$ can be read off the formula

$\displaystyle L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$

Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$$\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$$T_\zeta$ is the $E_\infty$$E_\infty$ cone over $\zeta$$\zeta$ and $TS^0$$TS^0$ is the free $E_\infty$$E_\infty$ spectrum generated by the sphere. In particular, its $\theta$$\theta$-algebra structure is free (see [Laures2003a]).

### 4.3 K(n)-homology computations

For Morava $K=K(n)$$K=K(n)$ at $p=2$$p=2$ one has an exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004a])

$\displaystyle \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$

which is induced by the obvious geometric maps. For $n=2$$n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see [Kitchloo&Laures&Wilson2004b])

$\displaystyle \xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$

### 4.4 Computations with respect to general complex oriented theories

Ando, Hopkins and Strickland investigated the homology ring $E_*BString$$E_*BString$ for even periodic multiplicative cohomology theories $E$$E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$$E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$$E_*BString$ is in terms of formal group data.

In [Ando&Hopkins&Strickland2001a] first the analogous complex problem is studied. The group $O\left< 8 \right>=String$$O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$$U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$$SU$. Consider the map

$\displaystyle (1-L_1)(1-L_2)(1-L_3): \xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$

where the $L_i$$L_i$ are the canonical line bundles over the individual factors. Since the first two Chern classes of the virtual bundle vanish the map lifts to $BU\left<6 \right>$$BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$$f$ in the cohomology ring

$\displaystyle (E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty )\cong E_0BU\left<6 \right> [\![x_1,x_2,x_3]\!]$

with $c_i(L_i)=x_i$$c_i(L_i)=x_i$. The power series $f$$f$ satisfies the following identities:

\displaystyle \begin{aligned} f(0,0,0)&=1\\ f(x_1,x_2,x_3)&=f(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)});\; \sigma \in \Sigma_3\\ f(x_1,x_2,x_3)f(x_0,x_1+_Fx_2,x_3)&=f(x_0+_Fx_1,x_2,x_3)f(x_0,x_1,x_2). \end{aligned}

Hence, the power series is an example a symmetric 3-cocycle or a cubical structure on the formal group law. In fact the main result of [Ando&Hopkins&Strickland2001a] is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$$E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$$f$ subject to the relations given by the 3 equations above.

The real version of this result has not been published yet by the three authors. Using the diagram

$\displaystyle \xymatrix{K{\mathbb Z},3)\ar[r]\ar[d]^=&BU\left< 6 \right>\ar[r]\ar[d]& BSU\ar[d]\\ K({\mathbb Z},3)\ar[r]& BString \ar[r]&BSpin}$

and the results for $K(2)_*BString$$K(2)_*BString$ described above they conjecture that $E_0BString$$E_0BString$ is the same quotient subject to the additional relation

$\displaystyle f(x_1,x_2,x_3)=1 \mbox{ for } x_1+_Fx_2+_Fx_3=0.$

## 5 The structure of the spectrum

Localized at a prime $p>3$$p>3$, string bordism splits additively into a sum of suspensions of $BP$$BP$, although the ring structure is different (see [Hovey2008]). For $p=3$$p=3$ there is a spectrum $Y$$Y$ with 3 cells in even dimensions such that $MString\wedge Y$$MString\wedge Y$ splits into a sum of suspensions of $BP$$BP$. For $p=2$$p=2$ it is hoped that the spectrum $tmf$$tmf$ splits off which is explained below.

Localized at a prime $p>3$$p>3$, the string bordism ring injects as a nonpolynomial subring of the oriented bordism ring. (A toy model worth bearing in mind is the inclusion
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$\mathbf{Z}[5x,y,xy] \into \mathbf{Z}[x,y]$.) Thus, Pontrjagin numbers suffice to distinguish elements of $\pi_*MString_{(p)}$$\pi_*MString_{(p)}$. In fact, a set $S$$S$ generates $\pi_*MString_{(p)}$$\pi_*MString_{(p)}$ as a $\mathbf{Z}_{(p)}$$\mathbf{Z}_{(p)}$-algebra if:
1. For each integer $n>1$$n>1$, there is an element $M^{4n}$$M^{4n}$ of $S$$S$ such that:
$\displaystyle \mathrm{ord}_p \big( s_n[M^{4n}] \big) = \begin{cases} 1 & \text{if 2n=p^i-1 or 2n=p^i+p^j for some integers 0 \le i \le j} \\ 0 & \text{otherwise} \end{cases}$
2. For each pair of integers $0$0, there is an element $N^{2(p^i+p^j)}$$N^{2(p^i+p^j)}$ of $S$$S$ such that:
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where $s_n$$s_n$ is the Milnor number, the characteristic number corresponding to the power sum polynomial of the Pontrjagin roots $\sum r_i^n$$\sum r_i^n$, and $s_{n_1,n_2}$$s_{n_1,n_2}$ is the characteristic number corresponding to the symmetric polynomial $\sum r_i^{n_1}r_j^{n_2}$$\sum r_i^{n_1}r_j^{n_2}$ (see [McTague2014, Theorem 4]).

## 6 The Witten genus

At the end of the 80s Ed Witten was studying the $S^1$$S^1$-equivariant index of the Dirac operator on a loop space of a $4k$$4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$$LM$ this would mean that $M$$M$ is $String$$String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$$k$. Nowadays this is called the Witten genus (see [Segal1988].) The Witten genus can be refined to a map of structured ring spectra

$\displaystyle W: MString \longrightarrow TMF$

from the Thom spectrum of String bordism to the spectrum $TMF$$TMF$ of topological modular forms ([Hopkins2002]). This map is also called the $\sigma$$\sigma$-orientation and is 15-connected (see [Hill2008]). The spectrum $TMF$$TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$$String$-bordism as $KO$$KO$-theory does for $Spin$$Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$$W$ gives characteristic numbers which together with $KO$$KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$$String$ bordism class. Moreover, $tmf$$tmf$ is supposed to be a direct summand of $MString$$MString$ as the orientation map $W$$W$ is shown to be surjective in homotopy (see [Hopkins&Mahowald2002].)