| Russian Mathematical Surveys |
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Brief communications The topology of the complement to the caustic of a Lagrangian germ of type V. D. Sedykh National State University of Oil and Gas "Gubkin University"
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     Light caustics, the evolutes of plane curves, and other envelopes of systems of rays are sets of critical values of so-called Lagrangian maps (see [1] and [5]). These sets are called caustics. By Arnold’s theorem on Lagrangian singularities, the germs of generic Lagrangian maps $f\colon L\to V$ of a smooth manifold $L$ into a smooth manifold $V$ of the same dimension $n\leqslant 5$ are stable and simple (that is, have zero modality). They are Lagrangian equivalent to the germs at the origin of the maps
$$
\begin{equation}
\mathbb{R}^n\to \mathbb{R}^n,\quad (\overline{t},\overline{q})\mapsto \biggl(-\frac{\partial S(\overline{t},\overline{q})} {\partial \overline{t}}\,,\overline{q}\biggr),\quad \overline{t}=(t_1,\dots,t_k),\quad \overline{q}=(q_{k+1},\dots,q_n),
\end{equation}
\tag{1}
$$
where $S=S(\overline{t},\overline{q})$ is a function of one of the following types ($\mu\leqslant n+1$ is an integer):
$$
\begin{equation*}
\begin{alignedat}{2} A_{\mu}^\pm\colon\ \ S&= \pm t_1^{\mu+1}+q_{\mu-1}t_1^{\mu-1}+\cdots+q_2t_1^2, &\qquad \mu&\geqslant 1; \\ D_{\mu}^{\pm}\colon\ \ S&= t_1^2t_2\pm t_2^{\mu-1}+q_{\mu-1}t_2^{\mu-2}+\cdots+q_3t_2^2, &\qquad \mu&\geqslant 4; \\ E_6^\pm\colon\ \ S&=t_1^3\pm t_2^4+q_5t_1t_2^2+q_4t_1t_2+q_3t_2^2, &\qquad \mu&=6. \end{alignedat}
\end{equation*}
\notag
$$
The equivalence class of the germ of a Lagrangian map at a critical point with respect to Lagrangian equivalence is called a (Lagrangian) singularity. The type of the function $S$ determines the type of this singularity (of a germ). The number $\operatorname{codim}X_{\mu}=\mu-1$ is called the codimension of type $X_{\mu}$ singularity. In what follows $X_{\mu}^\delta$ denotes $X_{\mu}^+$ for $\delta=+1$ and $X_{\mu}^-$ for $\delta=-1$. If $\mu$ is even or $\mu=1$, then Lagrangian germs of types $A_{\mu}^{+}$ and $A_{\mu}^{-}$ are Lagrangian equivalent and are denoted by $A_{\mu}$. To each point $y$ in the target space $V$ of a generic proper Lagrangian map $f$ with simple stable singularities we can assign the unordered set of symbols from Arnold’s theorem that are the types of the germs of $f$ at the preimages of the point $y$. The formal commutative product $\mathcal{A}$ of these symbols is called the type of the multisingularity of $f$ at $y$ (or the type of a monosingularity if $y$ has only one preimage). If the preimage $f^{-1}(y)$ is empty, then $\mathcal{A}=\mathbf{1}$. The set $\mathcal{A}_f$ of points $y\in V$ at which the map $f$ has a multisingularity of type $\mathcal{A}=X_1\cdots X_p$ is a smooth submanifold in the ambient space $V$. It is called the manifold of multisingularities of type $\mathcal{A}$. Its codimension $\operatorname{codim}\mathcal{A}$ is equal to $\sum_{i=1}^{p}\operatorname{codim}X_i$. In [2] we studied the topology of the manifolds of multisingularities for Lagrangian germs of types $A_\mu^{\pm}$ and $D_\mu^{\pm}$ (for all $n$). In particular, it follows from Theorem 7.8 in that paper that the total number of connected components of the complement to the caustic of the map (1) with singularity of type $D_{\mu}^\delta$ at the origin is equal to: $(k^2+3k-2)/2$ if $\mu=2k$, $\delta=+1$; $(k^2+k)/2$ if $\mu=2k$, $\delta=-1$; and $(k^2+ 3k)/2$ if $\mu= 2k+ 1$, $\delta=\pm1$. Among these, $(k^2-3k+2)/2$, $(k^2-k)/2$, and $(k^2- k)/2$ components, respectively, are homotopy equivalent to a circle and the other components are contractible. In [3], [4], and [6] we studied the manifolds of multisingularities for a Lagrangian germ of type $E_6^{\pm}$ at points of its caustic as well as the complement to the image. This note describes the topology of the connected components of the complement to the caustic of a germ $E_6^{\pm}$ in its image. Theorem. Let the Lagrangian map $f$ be given by (1). Assume that it has a singularity of type $E_6^{\pm}$ at the origin. Then the complement to the caustic of $f$ has seven connected components: two connected components of the manifold of multisingularities of each type $A_1^2$, $A_1^4$, and $A_1^6$, and one component is the complement to the image. All of them are contractible except for one connected component of type $A_1^4$. This non-contractible component is homotopy equivalent to a circle $S^1$. Its inverse image under $f$ has three connected components. The restriction of $f$ to one of them is a two-sheeted covering; the restriction to each of the other two is a diffeomorphism. Consider a generic proper Lagrangian map $f\colon L\to V$ with simple stable singularities. Assume that it has a multisingularity of type $\mathcal{B}$ at a point $y\in V$, where $\operatorname{codim}\mathcal{B}=c$. Fix a neighbourhood $U$ of the origin $0$ in $\mathbb{R}^{c}$, and consider a smooth embedding $h\colon U\to V$ such that $h(0)=y$ and the submanifold $h(U)\subset V$ is transversal to the manifold $\mathcal{B}_f$ at $y$. Let $B_{\varepsilon}\subset \mathbb{R}^{c}$ denote the open $c$-dimensional ball of radius $\varepsilon>0$ centred at $0$. Then there is a positive number $\varepsilon_0=\varepsilon_0(f,y,h)$ such that for all types $\mathcal{A}$ and all $\varepsilon<\varepsilon_0$ the set $h(B_{\varepsilon})\cap \mathcal{A}_f$ is a smooth manifold and the equivalence class of this manifold under diffeomorphisms depends only on the types $\mathcal{A}$ and $\mathcal{B}$. We denote this manifold by $\Xi_{\mathcal{A}}(\mathcal{B})$. A multisingularity of type $\mathcal{B}$ is adjacent to a multisingularity of type ${\mathcal A}$ if $\mathcal{A}\ne\mathcal{B}$ and $\Xi_\mathcal{A}(\mathcal{B})\ne\varnothing$. The Euler characteristic $J_{\mathcal{A}}(\mathcal{B})$ of the manifold $\Xi_{\mathcal{A}}(\mathcal{B})$ is called the adjacency index of a multisingularity of type $\mathcal{B}$ to a multisingularity of type $\mathcal{A}$. The adjacency of a multisingularity of type $\mathcal{B}$ to a multisingularity of type $\mathcal{A}$ is said to be simple if all connected components of $\Xi_{\mathcal{A}}(\mathcal{B})$ are contractible. Otherwise, this adjacency is called complicated. From [3], [4], [6], and the above theorem, we obtain the following result. Corollary. The indices of all adjacencies of a monosingularity of type $E_6^\delta$ to multisingularities of a generic Lagrangian map are as follows: Remark. In [2] we found a system of relations between the adjacency indices of Lagrangian multisingularities (see [2], Theorem 5.2). The indices listed in the above corollary satisfy all these relations. Other results on the topological properties of Lagrangian maps can be found in the book [7]. 
Citation in format AMSBIB
\Bibitem{Sed23} |
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