Families of Legendrian submanifolds via generating families
Let \(J^1M\) be the \((2n+1)\)-dimensional \(1\)-jet space of an \(n\)-dimensional smooth manifold \(M\) that is closed or diffeomorphic to \(\mathbb R^n\) outside of a compact set. The \(1\)-jet space is equipped with the standard contact structure \(\xi\). If \(\Lambda\subset J^1M\) is an \(n\)-dimensional compact Legendrian submanifold, then the topology of the space \(\mathcal{L}\) of all such submanifolds is formed by taking the quotient of the function space of Legendrian embeddings by orientation-preserving self-diffeomorphisms of the domain. Since the \(1\)-jet of a function \(g:M\to\mathbb R\) is a Legendrian submanifold of \(J^1M\), a function \(f:M\times\mathbb R^N\to\mathbb R\) is a generating family if \(0\) is a regular value of the function \(\partial_\eta f:M\times\mathbb R^N\to\mathbb R^N\). If \(\Sigma_f=\{(x,\eta)\in M\times\mathbb R^N:\partial_\eta f(x,\eta)=0\}\) is the fiber critical set, then a generating family \(f\) defines a Legendrian submanifold \(\Lambda_f=\{(x,\partial_\eta f(x,\eta),f(x,\eta)):(x,\eta)\in\Sigma_f\}\) as the \(1\)-jet of \(f\) along \(\Sigma_f\). If \(\delta:M\times\mathbb R^N\times\mathbb R^N\to\mathbb R\),\ \((x,\eta,\widetilde\eta)\mapsto f(x,\widetilde\eta)-f(x,\eta)\) is the difference function on the fiber product of the domain of \(f\) with itself, then, since the critical points of \(\delta\) with positive critical values correspond to the Reeb chords of \(\Lambda_f\), the generating family homology with coefficients in \(\mathbb Z/2\) is defined as \(\text{GH}_k(f)=H_{N+1+k}(\delta^\omega,\delta^\varepsilon;\mathbb Z/2)\), where \(\omega\) is a number larger than any critical value of \(\delta\) and where there are no critical values of \(\delta\) in \((0,\varepsilon)\). In this paper, the authors investigate families of Legendrian submanifolds in \(1\)-jet spaces by applying the theory of families of generating family homologies. They show that if the Legendrian \(\Lambda\subset J^1M\) has a generating family \(f\) with generating family homology \(\text{GH}_*(f)\), \({\mathcal L}\) is the space of Legendrian embeddings in \(J^1M\), and \(\pi_m({\mathcal L};\Lambda,f)\) is the subgroup of \(\pi_m({\mathcal L},\Lambda)\) consisting of homotopy classes of \(m\)-spheres of Legendrians based at \(\Lambda\) that lift to \(m\)-spheres of generating families based at \(f\), then there exists a morphism \(\Psi:\pi_m({\mathcal L};\Lambda,f)\to\text{End}_{m-1}(\text{GH}_*(f))\). Moreover, the authors show that for the space of Legendrian submanifolds of \(\mathbb R^{2n+1}\) the morphism is nontrivial, and there exists an infinite family of Legendrian \(n\)-spheres in \(\mathbb R^{2n+1}\) such that for each sphere \(\Lambda\), there exists \(\alpha\in\pi_m({\mathcal L},\Lambda)\) which is contractible as a smooth loop of spheres but is not contractible in the space of Legendrian submanifolds.
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