Morse-Bott theory on posets and a homological Lusternik-Schnirelmann theorem (Q6584684)

This paper constructs a discrete version of Morse-Bott theory on posets and proves a Lusternik-Schnirelmann theorem for Morse-Bott functions on posets.\N\NCompared with Morse theory, Morse-Bott theory does not require each critical point to be isolated, hence the set of critical points could be a submanifold. In the discrete setting, a Morse matching \(\mathcal{M}\) (corresponding to a discrete Morse function) on a poset \(X\) need to be acyclic and the points not belonging to \(\mathcal{M}\) are called critical. In this paper, the concept of critical subposet is generalized to both critical points and the points that belong in some cycle (Definition 3.4). This subposet is called the \textit{chain recurrent set} \(\mathcal{R}\). Then, a natural equivalence relation between two points \(x, y\in \mathcal{R}\) is defined as \N\(x \sim y\) if they are in the same cycle. Each equivalence class is called a \textit{basic set}.\N\NGiven a matching \(\mathcal{M}\) on a finite poset \(X\), a function \(f:X \to \mathbb{R}\) is called a \textit{Morse-Bott function} if it is constant on each basic set and is a discrete Morse function away from the critical subposet (Definition 3.5). Furthermore, Theorem 3.1 shows an approach to constructing a Morse-Bott function on a finite graded poset.\N\NIn Section 4, Theorems 4.1 and 4.2 recover the fundamental theorems of Morse-Bott theory in the setting of\N\(X\) being a finite homologically admissible poset and \(f\) a Morse-Bott function. Also, Theorem 4.3 and Corollary 4.1 show a poset-version of the strong and weak Morse-Bott inequalities, respectively.\N\NIn Section 5, the homological chain category is introduced (Definitions 5.1 and 5.2). Moreover, Theorem 5.1 shows the relationship between the homological chain category of a homologically admissible poset \(X\) and the homological chain category of the basis sets of \(X\) as an inequality. Thus as a corollary, hccat\((X)\) is a lower bound for the number of critical points of a discrete Morse function \(f\) (Corollary 5.3).\N\NFinally, in Section 6, a useful example is computed to illustrate the concepts and theorems.