Homotopy classes of total foliations.(English)Zbl 1244.57049
A triple \(\left(\xi^1,\xi^2,\xi^3\right)\) of smooth transversely oriented plane fields on an (oriented, closed) \(3\)-manifold \(M\) is a total plane field if \(\xi^1(p)\cap\xi^2(p) \cap \xi^3(p)=0\) at any \(p\in M\). A total plane field is a foliation if each \(\xi^i\) is integrable.
Vanishing of Euler characteristics and second Stiefel-Whitney classes implies the existence of total plane fields on any closed, oriented \(3\)-manifold.D.Hardorp [“All compact orientable three dimensional manifolds admit total foliations”, Mem.Am.Math.Soc.233, 74 p. (1980;Zbl 0435.57005)] proved that each closed, orientable \(3\)-manifold carries a total foliation. MoreoverJ.W.Wood [“Foliations on 3-manifolds”, Ann.Math.(2) 89, 336–358 (1969;Zbl 0176.21402)] proved that each plane field on a closed \(3\)-manifold is homotopic to a foliation.
In the paper under review it is proven that each total plane field on a closed, oriented \(3\)-manifold is homotopic to a total foliation. (Using the Eliashberg-Thurston theorem this then also implies that any oriented plane field with Euler class zero is homotopic to positive and negative contact structures which form a bicontact structure.)
The basic idea of this long and involved paper is that homotopy classes of total plane fields are determined by (differenences of) spin structures and Hopf degrees. Namely there are natural bijections between the set of total plane fields, the set of orthonormal frames on \(TM\), and the set of continuous maps from \(M\) to \(SO(3)\). For two total plane fields \(\xi_0^i\) and \(\xi^i\) let \(\Phi\left(\xi_0^i,\xi^i\right):M\rightarrow SO(3)\) be the corresponding map. Because of \(\pi_1SO(3)=\mathbb Z/2\mathbb Z\) this yields an element \(s\left(\xi_0^i,\xi^i\right)\in Hom\left(\pi_1M,\mathbb Z/2\mathbb Z\right)=H^1\left(M,\mathbb Z/2\mathbb Z\right)\) which is the difference of spin structures. If \(s\left(\xi_0^i,\xi^i\right)=0\), then \(\Phi\left(\xi_0^i,\xi^i\right)\) admits a lift \(\widetilde{\Phi}\left(\xi_0^i,\xi^i\right):M\rightarrow Spin\left(3\right)\) whose mapping degree is \(H\left(\xi_0^i,\xi^i\right)\), the difference of Hopf degrees. Two total plane fields \(\xi_0^i\) and \(\xi^i\) are homotopic if and only if \(s\left(\xi_0^i,\xi^i\right)=H\left(\xi_0^i,\xi^i\right)=0\). Thus the proof of the main theorem is reduced to showing that one can find a total foliation for any spin structure and difference of Hopf degrees.
The basic building block of the authors construction are so called \({\mathcal{R}}\)-components. These are total foliations \({\mathcal{F}}^i\) of \(S^1\times D^2\) such that \({\mathcal{F}}^3\) is a thick Reeb component (i.e., with a trivially foliated neighborhood of the boundary torus) and such that \({\mathcal{F}}^1\) and \({\mathcal{F}}^2\) are generated by the kernel of \(dy-\chi\left(y\right)dt\) and \(dx-\chi\left(y\right)dt\), respectively, where \(\chi\) is such that \(0<\chi\left(x\right)<1\) if \(x\in\left[\frac{1}{2},\frac{3}{2}\right]\) and \(\chi\left(x\right)=0\) otherwise.
The authors give an alternative proof of Hardorp’s theorem which shows in particular that a total foliation \({\mathcal{F}}^i\) exists for each given spin structure and with the additional condition that the total foliation contains two unknotted \({\mathcal{R}}\)-components \(R_\pm\) which are \(\pm1\)-framed, respectively.
The result implying the main theorem is then that for each \(n\) one can find a total foliation \({\mathcal{F}}_n^i\) with \(H\left({\mathcal{F}}^i,{\mathcal{F}}_n^i\right)=n\) (and actually such that the above additional condition holds). The authors first solve this problem (with the additional condition) for a special total foliation \({\mathcal{R}}^i\) on \(S^3\), namely they find \({\mathcal{R}}_n^i\) for the total foliation \({\mathcal{R}}^i\) consisting of two \(\left(-1\right)\)-framed unknotted \({\mathcal{R}}\)-components. The desired total foliation \({\mathcal{F}}_n\) on \(M\) is then obtained by gluing \({\mathcal{F}}^i\) and \({\mathcal{R}}^i_n\) along the boundaries of the \({\mathcal{R}}\)-components.
Vanishing of Euler characteristics and second Stiefel-Whitney classes implies the existence of total plane fields on any closed, oriented \(3\)-manifold.D.Hardorp [“All compact orientable three dimensional manifolds admit total foliations”, Mem.Am.Math.Soc.233, 74 p. (1980;Zbl 0435.57005)] proved that each closed, orientable \(3\)-manifold carries a total foliation. MoreoverJ.W.Wood [“Foliations on 3-manifolds”, Ann.Math.(2) 89, 336–358 (1969;Zbl 0176.21402)] proved that each plane field on a closed \(3\)-manifold is homotopic to a foliation.
In the paper under review it is proven that each total plane field on a closed, oriented \(3\)-manifold is homotopic to a total foliation. (Using the Eliashberg-Thurston theorem this then also implies that any oriented plane field with Euler class zero is homotopic to positive and negative contact structures which form a bicontact structure.)
The basic idea of this long and involved paper is that homotopy classes of total plane fields are determined by (differenences of) spin structures and Hopf degrees. Namely there are natural bijections between the set of total plane fields, the set of orthonormal frames on \(TM\), and the set of continuous maps from \(M\) to \(SO(3)\). For two total plane fields \(\xi_0^i\) and \(\xi^i\) let \(\Phi\left(\xi_0^i,\xi^i\right):M\rightarrow SO(3)\) be the corresponding map. Because of \(\pi_1SO(3)=\mathbb Z/2\mathbb Z\) this yields an element \(s\left(\xi_0^i,\xi^i\right)\in Hom\left(\pi_1M,\mathbb Z/2\mathbb Z\right)=H^1\left(M,\mathbb Z/2\mathbb Z\right)\) which is the difference of spin structures. If \(s\left(\xi_0^i,\xi^i\right)=0\), then \(\Phi\left(\xi_0^i,\xi^i\right)\) admits a lift \(\widetilde{\Phi}\left(\xi_0^i,\xi^i\right):M\rightarrow Spin\left(3\right)\) whose mapping degree is \(H\left(\xi_0^i,\xi^i\right)\), the difference of Hopf degrees. Two total plane fields \(\xi_0^i\) and \(\xi^i\) are homotopic if and only if \(s\left(\xi_0^i,\xi^i\right)=H\left(\xi_0^i,\xi^i\right)=0\). Thus the proof of the main theorem is reduced to showing that one can find a total foliation for any spin structure and difference of Hopf degrees.
The basic building block of the authors construction are so called \({\mathcal{R}}\)-components. These are total foliations \({\mathcal{F}}^i\) of \(S^1\times D^2\) such that \({\mathcal{F}}^3\) is a thick Reeb component (i.e., with a trivially foliated neighborhood of the boundary torus) and such that \({\mathcal{F}}^1\) and \({\mathcal{F}}^2\) are generated by the kernel of \(dy-\chi\left(y\right)dt\) and \(dx-\chi\left(y\right)dt\), respectively, where \(\chi\) is such that \(0<\chi\left(x\right)<1\) if \(x\in\left[\frac{1}{2},\frac{3}{2}\right]\) and \(\chi\left(x\right)=0\) otherwise.
The authors give an alternative proof of Hardorp’s theorem which shows in particular that a total foliation \({\mathcal{F}}^i\) exists for each given spin structure and with the additional condition that the total foliation contains two unknotted \({\mathcal{R}}\)-components \(R_\pm\) which are \(\pm1\)-framed, respectively.
The result implying the main theorem is then that for each \(n\) one can find a total foliation \({\mathcal{F}}_n^i\) with \(H\left({\mathcal{F}}^i,{\mathcal{F}}_n^i\right)=n\) (and actually such that the above additional condition holds). The authors first solve this problem (with the additional condition) for a special total foliation \({\mathcal{R}}^i\) on \(S^3\), namely they find \({\mathcal{R}}_n^i\) for the total foliation \({\mathcal{R}}^i\) consisting of two \(\left(-1\right)\)-framed unknotted \({\mathcal{R}}\)-components. The desired total foliation \({\mathcal{F}}_n\) on \(M\) is then obtained by gluing \({\mathcal{F}}^i\) and \({\mathcal{R}}^i_n\) along the boundaries of the \({\mathcal{R}}\)-components.
Reviewer: Thilo Kuessner (Seoul)
MSC:
| 57R30 | Foliations in differential topology; geometric theory |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
