On characteristic numbers of 24 dimensional string manifolds

Abstract

In this paper, we study the Pontryagin numbers of 24 dimensional String manifolds. In particular, we find representatives of an integral basis of the String cobrodism group at dimension 24, based on the work of Mahowald and Hopkins (The structure of 24 dimensional manifolds having normal bundles which lift to BO[8], from “Recent progress in homotopy theory” (D. M. Davis, J. Morava, G. Nishida, W. S. Wilson, N. Yagita, editors), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 89-110, 2002), Borel and Hirzebruch (Am J Math 80: 459–538, 1958) and Wall (Ann Math 75:163–198, 1962). This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the 2-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins and Singer (J Differ Geom 70:329–452, 2005), and also the 3-primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.

Appendix A: Twisted A-hats and twisted signatures

Let M be a 4m dimensional oriented closed smooth manifold. There are two important characteristic numbers, namely the (twisted) A-hat genus and the (twisted) signature, which are the topological pillars of the Atiyah–Singer index theory.

Equip M with a Riemannian metric $g^{TM}$. Let $\nabla ^{TM}$ be the associated Levi-Civita connection on TM and $R^{TM}=(\nabla ^{TM})^{2}$ be the curvature of $\nabla ^{TM}$. $\nabla ^{TM}$ extends canonically to a Hermitian connection $\nabla ^{T_{{\mathbf {C}}}M}$ on $T_{{\mathbf {C}} }M=TM\otimes {\mathbf {C}}$, the complexification of TM.

Let ${\widehat{A}}(TM,\nabla ^{TM})$ be the Hirzebruch ${\widehat{A}}$-form defined by (cf. [32])

$$\begin{aligned} {\widehat{A}}(TM,\nabla ^{TM})={\det }^{1/2}\left( {\frac{{\frac{\sqrt{-1}}{ 4\pi }}R^{TM}}{\sinh \left( {\frac{\sqrt{-1}}{4\pi }}R^{TM}\right) }}\right) . \end{aligned}$$

(A.1)

Let E be a Hermitian vector bundles over M carrying a Hermitian connection $\nabla ^{E}$. Let $R^{E}=(\nabla ^{E})^{2}$ be the curvature of $\nabla ^{E}$. The Chern character form (cf. [32]) is defined as

$$\begin{aligned} \mathrm {ch}(E,\nabla ^{E})=\mathrm {tr}\left[ \exp \left( {\frac{\sqrt{-1}}{ 2\pi }}R^{E}\right) \right] . \end{aligned}$$

(A.2)

The ${\widehat{A}}$-genus and the twisted ${\widehat{A}}$-genus are defined respectively as

$$\begin{aligned} \begin{aligned}&{\widehat{A}}(M)=\int _M {\widehat{A}}(TM,\nabla ^{TM}), \\&{\widehat{A}}(M,E)=\int _M {\widehat{A}}(TM,\nabla ^{TM})\mathrm {ch}(E,\nabla ^{E}). \end{aligned} \end{aligned}$$

(A.3)

When M is spin, let $S(TM)=S_{+}(TM)\oplus S_{-}(TM)$ denote the bundle of complex spinors associated to the Spin structure. Then S(TM) carries induced Hermitian metric and connection preserving the above $\mathbf{Z}_2$-grading. Let

$$\begin{aligned} D_{\pm }:\Gamma (S_{\pm }(TM))\rightarrow \Gamma (S_{\mp }(TM)) \end{aligned}$$

denote the induced Spin Dirac operators (cf. [14]). By the Atiyah-Singer index theorem,

$$\begin{aligned} \begin{aligned}&{\widehat{A}}(M)=\mathrm {Ind}(D), \\&{\widehat{A}}(M,E)=\mathrm {Ind}(D\otimes E). \end{aligned} \end{aligned}$$

(A.4)

Let ${\widehat{L}}(TM,\nabla ^{TM})$ be the Hirzebruch characteristic form defined by (cf. [16, 32])

$$\begin{aligned} {\widehat{L}}(TM,\nabla ^{TM})={\det }^{1/2}\left( {\frac{{\frac{\sqrt{-1}}{ 2\pi }}R^{TM}}{\tanh \left( {\frac{\sqrt{-1}}{4\pi }}R^{TM}\right) }}\right) . \end{aligned}$$

(A.5)

Note that ${\widehat{L}}(TM,\nabla ^{TM})$ defined here is different from the classical Hirzebruch L-form defined by

$$\begin{aligned} L(TM,\nabla ^{TM})={\det }^{1/2}\left( {\frac{{\frac{\sqrt{-1}}{ 2\pi }}R^{TM}}{\tanh \left( {\frac{\sqrt{-1}}{2\pi }}R^{TM}\right) }}\right) . \end{aligned}$$

However they give same top (degree 4m) forms and therefore

$$\begin{aligned} \int _M{\widehat{L}}(TM,\nabla ^{TM}) =\int _M L(TM,\nabla ^{TM}). \end{aligned}$$

(A.6)

We would also like to point out that our ${\widehat{L}}$ is different from the $\mathbf{{\widehat{L}}}$ in page 233 of [14].

Let $\mathrm {ch}(E,\nabla ^{E})=\sum _{i=0}^{2m}\mathrm {ch}^{i}(E,\nabla ^{E})$ such that $\mathrm {ch}^{i}(E,\nabla ^{E})$ is the degree 2i component. Define

$$\begin{aligned} \mathrm {ch}_2(E,\nabla ^{E})=\sum _{i=0}^{2m}2^i\mathrm {ch}^{i}(E,\nabla ^{E}).\end{aligned}$$

(A.7)

It’s not hard to see that

$$\begin{aligned} \int _M {\widehat{L}}(TM,\nabla ^{TM}) \mathrm {ch}(E,\nabla ^{E})=\int _M L(TM,\nabla ^{TM}) \mathrm {ch}_2(E,\nabla ^{E}).\end{aligned}$$

(A.8)

Let $\Lambda _{\mathbb {C}}(T^*M)$ be the complexified exterior algebra bundle of TM. Let $\langle \ , \ \rangle _{\Lambda _{\mathbb {C}}(T^*M)}$ be the Hermitian metric on $\Lambda _{\mathbb {C}}(T^*M)$ induced by $g^{TM}$. Let dv be the Riemannian volume form associated to $g^{TM}$. Then $\Gamma (M, \Lambda _{\mathbb {C}}(T^*M))$ has a Hermitian metric such that for $\alpha , \alpha '\in \Gamma (M, \Lambda _{\mathbb {C}}(T^*M))$,

$$\begin{aligned} \langle \alpha , \alpha '\rangle =\int _{M}\langle \alpha , \alpha '\rangle _{\Lambda _{\mathbb {C}}(T^*M)}\,dv. \end{aligned}$$

For $X\in TM$, let c(X) be the Clifford action on $\Lambda _{\mathbb {C}}(T^*M)$ defined by $c(X)=X^*-i_X$, where $X^*\in T^*M$ corresponds to X via $g^{TM}$. Let $\{e_1,e_2, \cdots , e_{2n}\}$ be an oriented orthogonal basis of TM. Set

$$\begin{aligned} \Omega =(\sqrt{-1})^{n}c(e_1)\cdots c(e_{2n}). \end{aligned}$$

Then one can show that $\Omega$ is independent of the choice of the orthonormal basis and $\Omega _E=\Omega \otimes 1$ is a self-adjoint operator on $\Lambda _{\mathbb {C}}(T^*M)\otimes E$ such that $\Omega _E^2=\mathrm {Id}|_{\Lambda _{\mathbb {C}}(T^*M)\otimes E}$.

Let d be the exterior differentiation operator and $d^*$ be the formal adjoint of d with respect to the Hermitian metric. The operator

$$\begin{aligned} D_{Sig}:=d+d^*=\sum _{i=1}^{2n} c(e_i)\nabla _{e_i}^{\Lambda _{\mathbb {C}}(T^*M)}: \Gamma (M, \Lambda _{\mathbb {C}}(T^*M))\rightarrow \Gamma (M, \Lambda _{\mathbb {C}}(T^*M)) \end{aligned}$$

is the signature operator and the more general twisted signature operator is defined as (cf. [9])

$$\begin{aligned} D_{Sig}\otimes E:=\sum _{i=1}^{2n} c(e_i)\nabla _{e_i}^{\Lambda _{\mathbb {C}}(T^*M)\otimes E}: \Gamma (M, \Lambda _{\mathbb {C}}(T^*M)\otimes E)\rightarrow \Gamma (M, \Lambda _{\mathbb {C}}(T^*M)\otimes E). \end{aligned}$$

The operators $D_{Sig}\otimes E$ and $\Omega _E$ are anti-commutative. If we decompose $\Lambda _{\mathbb {C}}(T^*M)\otimes E=\Lambda ^+_{\mathbb {C}}(T^*M)\otimes E\oplus \Lambda ^-_{\mathbb {C}}(T^*M)\otimes E$ into $\pm 1$ eigenspaces of $\Omega _E$, then $D_{Sig}\otimes E$ decomposes to define

$$\begin{aligned} (D_{Sig}\otimes E)^{\pm }: \Gamma (M, \Lambda ^{\pm }_{\mathbb {C}}(T^*M)\otimes E)\rightarrow \Gamma (M, \Lambda ^{\mp }_{\mathbb {C}}(T^*M)\otimes E).\end{aligned}$$

(A.9)

The twisted signature of M is defined as the index of the operator $(D_{Sig}\otimes E)^{+}$ denoted by $\mathrm {Sig}(M, E)$,

$$\begin{aligned} \mathrm {Sig}(M, E)=\mathrm {Ind}((D_{Sig}\otimes E)^{+}).\end{aligned}$$

(A.10)

By the Atiyah-Singer index theorem,

$$\begin{aligned} \mathrm {Sig}(M, E)=\int _M {\widehat{L}}(TM,\nabla ^{TM})\mathrm {ch}(E, \nabla ^E). \end{aligned}$$

Note that in the book [14] (Theorem 13.9), the following formula is given

$$\begin{aligned} \mathrm {Sig}(M, E)=\int _M L(TM,\nabla ^{TM}) \mathrm {ch}_2(E,\nabla ^{E}). \end{aligned}$$

There is an important twisted ${\widehat{A}}$-genus, namely the Witten genus [30] by coupling ${\widehat{A}}(M)$ with the Witten bundle [30]

$$\begin{aligned} \Theta (T_{{\mathbb {C}}}M)=\overset{\infty }{\underset{n=1}{\otimes }} S_{q^{2n}}(\widetilde{T_{{\mathbb {C}}}M}),\ \ \mathrm{with}\ \ \widetilde{T_{{\mathbb {C}}}M}=TM\otimes {\mathbb {C}}-{{\mathbb {C}}}^{4m}. \end{aligned}$$

The Witten genus then can defined as

$$\begin{aligned} W(M)=\left\langle {\widehat{A}}(TM)\mathrm {ch}\left( \Theta \left( T_{\mathbb {C }}M\right) \right) ,[M]\right\rangle . \end{aligned}$$