Russian Mathematical Surveys, 2022, Volume 77, Issue 5, Pages 943–945 DOI: https://doi.org/10.4213/rm10032e (Mi rm10032)

DOI: https://doi.org/10.4213/rm10032e

Funding agency	Grant number
HSE Basic Research Program
Ministry of Education and Science of the Russian Federation	5-100
Russian Foundation for Basic Research	18-01-00461
Simons Foundation
The research of M. S. Temkin was carried out within the framework of the Basic Research Program at the HSE University and funded by the Russian Academic Excellence Project “5-100”. The work of P. S. Pushkar was supported by the Simons Foundation.

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Consider a closed smooth manifold $M$ and a Morse function $f$ on it. Choosing a generic Riemannian metric (and suitable orientations), one can define a Morse complex which is formally based on the critical points of the function $f$. Assume that all critical values of $f$ are different (such a function is called strict). Then there is a linear order on the basis elements, and therefore the matrix of the differential is defined. This matrix depends on the Riemannian metric. In this note we formulate some restrictions on the form of the differential matrix in terms of the function $f$. For this purpose we introduce an invariant of a function under continuous deformations in the (disconnected) space of strict Morse functions on a fixed manifold. This invariant is an enhancement of the Barannikov pairs [1].

Fix a strict Morse function $f$. Given $a \in \mathbb{R}$, define the set $M^a:=\{x\in M$: $f(x) \leqslant a\}$, called the sublevel set. For each critical point $x \in M$ choose a generator of the group $\mathsf{H}_{\operatorname{ind}x}(M^{f(x)+\varepsilon}, M^{f(x)-\varepsilon};\mathbb{Z}) \simeq \mathbb{Z}$, where $\operatorname{ind}x$ is the index of $x$ and $\varepsilon \in \mathbb{R}_+$ is small enough. This choice is called an orientation of $f$. Denote by $\operatorname{Cr}_k \subset M$ the set of critical points of index $k$. Then we obtain the matrix of the $k$th differential in the Morse complex. This matrix depends on the generic Riemannian metric and has size $\mathbin{\#} \operatorname{Cr}_{k-1}\times\mathbin{\#}\operatorname{Cr}_k$ (for $k \in\{1,\dots,\dim M\}$). Recall that the $(i,j)$th entry of the matrix is equal to the number of (non-parametrized) antigradient trajectories from the $j$th point of index $k$ to the $i$th point of index $k-1$, taken with appropriate signs. Here we number the points of index $d$ from $1$ to $\# \operatorname{Cr}_d$ in the ascending order of critical values, using the fact that $f$ is strict. Therefore, each critical point has two parameters: the index and the order number.

We turn to the description of the invariant of an oriented strict Morse function $f$. (The conditions on the matrix of the differential will be formulated below in terms of this invariant.) The invariant consists of two parts, the Barannikov pairs (or simply pairs) and the Bruhat numbers. A Barannikov pair is a pair $(x,y)$ of critical points of $f$ with consecutive indices which is subject to the condition: if $f(x) > f(y)$, then $\operatorname{ind}x=\operatorname{ind} y+1$. Each critical point can belong to at most one Barannikov pair. The conditions above are necessary but not sufficient; the construction is presented below. Now, the Bruhat number is a non-zero rational number assigned to each Barannikov pair. An example is shown in Fig. 1. Critical points are shown as dots ordered from bottom to top in the ascending order of critical values. The index is indicated above or below the dot, and pairs are indicated by line segments. The Bruhat number of a pair is written to the left of the corresponding segment.

Zoom inZoom out

Figure 1

Now we present the construction of the invariant. In what follows, homology is taken with coefficients in $\mathbb{Q}$. Let $x$ and $y$ be critical points of index $k+1$ and $k$, respectively, such that $f(x) > f(y)$. By the fundamental theorem of Morse theory there is a homotopy equivalence $M^{f(x)+\varepsilon} \simeq M^{f(x)-\varepsilon} \cup_\varphi e^{k+1}$, where $e^{k+1}$ is a cell of dimension $k+1$ with characteristic map $\varphi \colon S^k \to M^{f(x)-\varepsilon}$, which can be assumed to be an embedding. The fundamental class of the corresponding sphere can be viewed as an element of $\mathsf{H}_k(M^{f(x)-\varepsilon})$. Let $X$ be the image of this class under the map $\mathsf{H}_k(M^{f(x)-\varepsilon}) \to \mathsf{H}_k(M^{f(x)-\varepsilon}, M^{f(y)-\varepsilon})$. Next, the fundamental class of the cell corresponding to the point $y$ can be viewed as an element of $\mathsf{H}_k(M^{f(y)+\varepsilon},M^{f(y)-\varepsilon})$. Let $Y$ be the image of this class under the map $\mathsf{H}_k(M^{f(y)+\varepsilon},M^{f(y)-\varepsilon}) \to \mathsf{H}_k(M^{f(x)-\varepsilon},M^{f(y)-\varepsilon})$ induced by the embedding. The points $x$ and $y$ form a Barannikov pair if and only if $X=\lambda Y \ne 0$ for some $\lambda \in \mathbb{Q}$. The number $\lambda$ is called the Bruhat number of the corresponding pair. When the orientation of $f$ is reversed, some Bruhat numbers can change sign.

The equivalence of this definition of Barannikov pairs and the original definition [1] was shown in [3]; Bruhat numbers were also introduced there. Moreover, in [3] Bruhat numbers can take values in any field, not only in $\mathbb{Q}$. For the present note, however, this generality is irrelevant. Note that similar ideas about Bruhat numbers over $\mathbb{Q}$ appeared independently in [2].

For fixed $k$ we define the following subset $\mathcal{T}$ of all integer matrices of size $\mathbin{\#}\operatorname{Cr}_{k-1}\times\mathbin{\#}\operatorname{Cr}_k$. We say that a cell $(i,j)$ is covered if there exists a pair of indices $(i',j')$ such that: 1) the points with numbers $i'$ and $j'$ form a Barannikov pair; 2) $(i < i' \text{ AND } j \geqslant j') \text{ OR } (i \leqslant i' \text{ AND } j > j' )$. Then a matrix $M$ belongs to $\mathcal{T}$ if and only if the following conditions are satisfied: 1) if a cell $(i,j)$ is not covered and the points $i$ and $j$ do not form a Barannikov pair, then $M_{i,j}=0$; 2) if a cell $(i,j)$ is not covered and the points $i$ and $j$ form a Barannikov pair, then the entry $M_{i,j}$ is equal to the Bruhat number of this pair. In the example in Fig. 1 for $k=2$ the set $\mathcal{T}$ consists of matrices of the form $\begin{pmatrix} * & * & * \\ 3 & * & * \\ 0 & * & * \\ 0 & 2 & * \\ \end{pmatrix}$.

Bibliography
1.	S. A. Barannikov, Singularities and bifurcations, Adv. Soviet Math., 21, 1994, 93–115
2.	D. Le Peutrec, F. Nier, and C. Viterbo, Bar codes of persistent cohomology and Arrhenius law for $p$-forms, 2020, 147 pp., arXiv: 2002.06949
3.	P. Pushkar and M. Tyomkin (Temkin), “Enhanced Bruhat decomposition and Morse theory”, Int. Math. Res. Not. IMRN (to appear); 2021 (v1 – 2020), 53 pp., arXiv: 2012.05307

Citation in format AMSBIB

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\by P.~E.~Pushkar', M.~S.~Temkin
\paper On the differential matrix in the Morse complex
\jour Russian Math. Surveys
\yr 2022
\vol 77
\issue 5
\pages 943--945
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