String bordism

From Manifold Atlas
(Difference between revisions)
Jump to: navigation, search
(The Witten genus)
m (The bordism groups)
(25 intermediate revisions by 5 users not shown)
Line 13: Line 13:
== The String group ==
== The String group ==
<wikitex>;
<wikitex>;
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 $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2010}). A more explicit model for this extension can be found in \cite{Meinrecken03}.
+
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 $String(n)$ which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2009}). A more explicit model for this extension can be found in \cite{Meinrenken2003}.
</wikitex>
</wikitex>
== The bordism groups ==
== The bordism groups ==
<wikitex>;
<wikitex>;
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey97}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.
+
The additive structure of the bordism groups is not fully determined yet. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}) and the 3 torsion is annihalated by multiplication with 3 (see \cite{Hovey1997}). Moreover, the bordism groups $\Omega_{k}^{String}$ are finite for $k=1,2,3$ mod 4.
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:
Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. The first 16 bordism groups have been computed by Giambalvo \cite{Giambalvo1971|p. 538}:
Line 27: Line 27:
* $\Omega_{11}^{String} = 0$.
* $\Omega_{11}^{String} = 0$.
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.
* $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.
* $\Omega_{13}^{String} = 0$.
+
* $\Omega_{13}^{String} \cong \Zz/3$, generated by an [[Exotic spheres|exotic 13-sphere]].
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].
* $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]].
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]].
+
* $\Omega_{15}^{String} \cong \Zz/2$, genreated by an [[Exotic spheres|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 \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}.
At the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}.
Line 37: Line 37:
===Singular homology===
===Singular homology===
<wikitex>;
<wikitex>;
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong63}:
+
The cohomology ring $H^*(BString,{\mathbb Z}/p)$ has been computed for $p=2$ by Stong in \cite{Stong1963}:
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i|\sigma_2(i+1)>4]\otimes H^*(K({\mathbb Z},4))/Sq^2(\iota).$$
+
$$ H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 4]\otimes H^*(K({\mathbb Z},8))/Sq^2(\iota).$$
Here, $\sigma_2$ is the number of digits in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables. For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo69}.
+
Here, $\sigma_2$ is the number of ones in the duadic decomposition and the $\theta_i$ come from the cohomology of $BO$ and coincide with the Stiefel-Whitney up to decomposables.
+
+
From Stong's proof it follows that $ H^*(BO)\to H^*(BString)$ is surjective, and $H^*(BString)\cong {\mathbb Z}/2[\theta_i\mid \sigma_2(i-1)\ge 3]$ is a polynomial algebra.
+
+
For odd $p$ the corresponding result has been obtained by Giambalvo \cite{Giambalvo1969}.
</wikitex>
</wikitex>
Line 47: Line 51:
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula
$K(1)$ locally $MString$ coincides with $MSpin$ and decomposes into a wedge of copies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$
$$ L_{K(1)}MString \cong T_\zeta \wedge \bigwedge TS^0.$$
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Lau03}).
+
Here, $\zeta\in \pi_{-1}L_{K(1)}S^0$ is a generator, $T_\zeta$ is the $E_\infty $ cone over $\zeta$ and $TS^0$ is the free $E_\infty$ spectrum generated by the sphere. In particular, its $\theta$-algebra structure is free (see \cite{Laures2003a}).
</wikitex>
</wikitex>
===K(n)-local computations===
+
===K(n)-homology computations===
<wikitex>;
<wikitex>;
For $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{KLW04a})
+
For Morava $K=K(n)$ at $p=2$ one has an exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004a})
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$
$$ \xymatrix{K_* \ar[r] & K_*K({\mathbb Z}/2,2)\ar[r]& K_*K({\mathbb Z},3)\ar[r]& K_*BString \ar[r]&K_*BSpin \ar[r]& K_* K({\mathbb Z},4)\ar[r]^2& K({\mathbb Z},4)\ar[r]&K_*}$$
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{KLW04b})
+
which is induced by the obvious geometric maps. For $n=2$ it algebraically reduces to the split exact sequence of Hopf algebras (see \cite{Kitchloo&Laures&Wilson2004b})
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$
$$\xymatrix{K(2)_* \ar[r] & K(2)_*K({\mathbb Z},3)\ar[r]& K(2)_*BString \ar[r]&K(2)_*BSpin \ar[r]& K(2)_*}.$$
</wikitex>
</wikitex>
Line 64: Line 68:
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data.
Ando, Hopkins and Strickland investigated the homology ring $E_*BString$ for even periodic multiplicative cohomology theories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BString$ is in terms of formal group data.
In \cite{AHS2001} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map
+
In \cite{Ando&Hopkins&Strickland2001a} first the analogous complex problem is studied. The group $O\left< 8 \right>=String$ has a complex relative $U\left< 6 \right>$ which is defined in the same way by killing the third homotopy group of $SU$. Consider the map
$$ (1-L_1)(1-L_2)(1-L_3):
$$ (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}$$
\xymatrix{ {\mathbb C}P^\infty \times {\mathbb C}P^\infty\times {\mathbb C}P^\infty\ar[r]& BU}$$
Line 77: Line 81:
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).
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}$$
\end{aligned}$$
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 \cite{AHS2001} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ subject to the relations given by the 3 equations above.
+
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 \cite{Ando&Hopkins&Strickland2001a} is that it is the universal example of such a structure. Explicitly, this means that the commutative ring $E_0 BU\left <6 \right>$ is freely generated by the coefficients of $f$ 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
The real version of this result has not been published yet by the three authors. Using the diagram
Line 88: Line 92:
== The structure of the spectrum==
== The structure of the spectrum==
<wikitex>;
<wikitex>;
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below.
+
Localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). For $p=3$ there is a spectrum $Y$ with 3 cells in even dimensions such that $MString\wedge Y$ splits into a sum of suspensions of $BP$. For $p=2$ it is hoped that the spectrum $tmf$ splits off which is explained below.
+
Localized at a prime $p>3$, the string bordism ring injects as a nonpolynomial subring of the oriented bordism ring. (A toy model worth bearing in mind is the inclusion $\mathbf{Z}[5x,y,xy] \into \mathbf{Z}[x,y]$.) Thus, Pontrjagin numbers suffice to distinguish elements of $\pi_*MString_{(p)}$. In fact, a set $S$ generates $\pi_*MString_{(p)}$ as a $\mathbf{Z}_{(p)}$-algebra if:
+
<ol><li>For each integer $n>1$, there is an element $M^{4n}$ of $S$ such that:
+
$$
+
\mathrm{ord}_p \big( s_n[M^{4n}] \big) =
+
\begin{cases}
+
1 & \text{if $2n=p^i-1$ or $2n=p^i+p^j$ for some integers $0 \le i \le j$} \\
+
0 & \text{otherwise}
+
\end{cases}
+
$$</li>
+
<li>For each pair of integers $0<i<j$, there is an element $N^{2(p^i+p^j)}$ of $S$ such that:
+
$$\begin{align*}
+
s_{(p^i+p^j)/2}[N^{2(p^i+p^j)}]&=0
+
&\text{but}&&
+
s_{(p^i+1)/2,(p^j-1)/2}[N^{2(p^i+p^j)}] &\not\equiv 0 \mod p^2
+
\end{align*}$$where $s_n$ is the Milnor number, the characteristic number corresponding to the power sum polynomial of the Pontrjagin roots $\sum r_i^n$, and $s_{n_1,n_2}$ is the characteristic number corresponding to the symmetric polynomial $\sum r_i^{n_1}r_j^{n_2}$ (see {{cite|McTague2014|Theorem 4}}).
+
</li></ol>
</wikitex>
</wikitex>
== The Witten genus ==
== The Witten genus ==
<wikitex>;
<wikitex>;
At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)
+
At the end of the 80s Ed Witten was studying the $S^1$-equivariant index of the Dirac operator on a loop space of a $4k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.)
The Witten genus can be refined to a map of structured ring spectra
The Witten genus can be refined to a map of structured ring spectra
$$W: MString \longrightarrow TMF$$
$$W: MString \longrightarrow TMF$$
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller (see \cite{Hopkins2002}.) It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)
+
from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Hopkins2002}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.)
</wikitex>
</wikitex>

Latest revision as of 10:53, 13 July 2017

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 1 Introduction

String-bordism or 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> is the homotopy fibre of the map from BSpin given by half of the first Pontryagin class. The name String-group is due to Haynes Miller and will be explained below.


[edit] 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) which involves only finite dimensional manifolds is the following: the String group fibers over the Spin group with fibre K({\mathbb Z},2). One may think of K({\mathbb Z},2) as the realization of S^1 viewed as a smooth category with only one object. This way, the A_\infty space String(n) appears as the realization of a smooth 2-group extension of Spin(n) by the finite dimensional Lie groupoid S^1 (see [Schommer-Pries2009]). A more explicit model for this extension can be found in [Meinrenken2003].

[edit] 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} are finite for k=1,2,3 mod 4.

Clearly, since 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_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, generated by exotic 9-spheres.
  • \Omega_{10}^{String} \cong \Zz/6, generated by an exotic 10-sphere.
  • \Omega_{11}^{String} = 0.
  • \Omega_{12}^{String} \cong \Zz, generated by a 5-connected manifold with signature 8 \times 992.
  • \Omega_{13}^{String} \cong \Zz/3, generated by an exotic 13-sphere.
  • \Omega_{14}^{String} \cong \Zz/2, generated by the exotic 14-sphere.
  • \Omega_{15}^{String} \cong \Zz/2, genreated by an exotic 15-sphere.
  • \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].

[edit] 4 Homology calculations

[edit] 4.1 Singular homology

The cohomology ring H^*(BString,{\mathbb Z}/p) has been computed for 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 is the number of ones in the duadic decomposition and the \theta_i come from the cohomology of BO and coincide with the Stiefel-Whitney up to decomposables.

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

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


[edit] 4.2 K(1)-local computations

K(1) locally MString coincides with MSpin and decomposes into a wedge of copies of KO. However, it is not an algebra over KO. Its multiplicative structure for 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 is a generator, T_\zeta is the E_\infty cone over \zeta and TS^0 is the free E_\infty spectrum generated by the sphere. In particular, its \theta-algebra structure is free (see [Laures2003a]).


[edit] 4.3 K(n)-homology computations

For Morava K=K(n) at 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 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)_*}.

[edit] 4.4 Computations with respect to general complex oriented theories

Ando, Hopkins and Strickland investigated the homology ring E_*BString for even periodic multiplicative cohomology theories E. Even periodic theories are complex orientable which means that E^0({\mathbb C}P^\infty) carries a formal group. The description of 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 has a complex relative U\left< 6 \right> which is defined in the same way by killing the third homotopy group of 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 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>. If we choose a complex orienatation the lift gives a class 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. The power series 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> is freely generated by the coefficients of 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 described above they conjecture that 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.

[edit] 5 The structure of the spectrum

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

Localized at a prime 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
Tex syntax error
.) Thus, Pontrjagin numbers suffice to distinguish elements of \pi_*MString_{(p)}. In fact, a set S generates \pi_*MString_{(p)} as a \mathbf{Z}_{(p)}-algebra if:
  1. For each integer n>1, there is an element M^{4n} of 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<i<j, there is an element N^{2(p^i+p^j)} of S such that:
    Tex syntax error
    where s_n is the Milnor number, the characteristic number corresponding to the power sum polynomial of the Pontrjagin roots \sum r_i^n, and s_{n_1,n_2} is the characteristic number corresponding to the symmetric polynomial \sum r_i^{n_1}r_j^{n_2} (see [McTague2014, Theorem 4]).

[edit] 6 The Witten genus

At the end of the 80s Ed Witten was studying the S^1-equivariant index of the Dirac operator on a loop space of a 4k-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space LM this would mean that M is 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. 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 of topological modular forms ([Hopkins2002]). This map is also called the \sigma-orientation and is 15-connected (see [Hill2008]). The spectrum TMF was developed by Goerss, Hopkins and Miller. It is supposed to play the same role for String-bordism as KO-theory does for Spin-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map W gives characteristic numbers which together with KO and Stiefel-Whitney numbers are conjectured to determine the String bordism class. Moreover, tmf is supposed to be a direct summand of MString as the orientation map W is shown to be surjective in homotopy (see [Hopkins&Mahowald2002].)

[edit] 7 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox