For a smooth closed \(d\)-dimensional manifold \(M^d\), the author discusses the definition of \(k\)-width of \(M^d\), defined as the minimax of a numerical property \(P\) for preimages \(\pi^{-1}(x)\), where the minimum is taken over an appropriate class of maps \(\pi: M^d \to \mathbb{R}^k\) and the maximum is taken over all regular values \(x \in \mathbb{R}^k\). Further, the author defines \(k\)-width of an embedding \(e: M^d \to \mathbb{R}^k\) as the minimax of a numerical property \(P\) for preimages \((\pi\circ e)^{-1}(x)\). The main results are as follow. For any orientable 3-manifold \(M\), 2-width of \(M\) is equal to or smaller than two. Further, the author determines \(M\) whose 2-width is one. However, for 2-width of embeddings, there is a family of embeddings \(T^3 \to \mathbb{R}^4\) for which 2-width diverges.
The motivation of this paper is to investigate the genus \(g\) Goeritz group \(G_g\) for \(g \geq 4\); whether \(G_g\) is finitely generated or infinitely generated. Here, \(G_g\) is the fundamental group of the space of genus \(g\) Heegaard surfaces in \(S^3\). The author discusses the 2-width called frame complexity and gives a necessary and sufficient condition that \(G_g\)(\(g \geq4\)) is infinitely generated.