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A Morse function on the torus is perfect over every field

Example

Assume ${A C}_{ω}$ . On the two-dimensional torus $T^{2} = (R / Z)^{2}$ (The two-dimensional torus $T^{2} = (R / Z)^{2}$ ) let $f (x, y) = \cos (2 π x) + \cos (2 π y) .$ Its critical points are the four points with coordinates in ${0, \frac{1}{2}}$ , with Hessians $diag (- 4 π^{2} \cos (2 π x), - 4 π^{2} \cos (2 π y))$ , so the indices are $2$ at $(0, 0)$ , $1$ at $(0, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ , and $0$ at $(\frac{1}{2}, \frac{1}{2})$ : $M_{f} (t) = 1 + 2 t + t^{2}$ . The index-ordered handle presentation has one $0$ -handle, two $1$ -handles and one $2$ -handle, and its handle chain complex over any field $F$ has ranks $1, 2, 1$ ; the boundary coefficients are the intersection numbers of attaching and belt spheres: each $1$ -handle attaches with two feet on the belt circle of the $0$ -handle and the attaching circle of the $2$ -handle meets each belt $0$ -sphere of a $1$ -handle in two points, and in both cases the two contributions have opposite signs (and cancel mod two), so $\partial_{1} = \partial_{2} = 0$ . Hence $H_{0} ≅ F$ , $H_{1} ≅ F^{2}$ , $H_{2} ≅ F$ and $P_{T^{2}, F} (t) = 1 + 2 t + t^{2} = M_{f} (t)$ : $f$ is $F$ -perfect for every field, and the Euler identity gives $1 - 2 + 1 = 0 = χ (T^{2})$ .

Facts & Assumptions

Given: The flat torus $T^{2} = (R / Z)^{2}$ , the function $f (x, y) = \cos (2 π x) + \cos (2 π y)$ , and a field $F$ .

[F1]

Critical points, nondegeneracy, the Hessian and the index are the local Morse notions of the library (Critical points and critical values of a smooth function, Nondegenerate critical points, nullity, index, and coindex, The intrinsic Hessian of a smooth function at a critical point).

[F2]

The Morse numbers and Morse polynomial are $m_{k} (f) = # {p : ind (p) = k}$ and $M_{f} (t) = \sum_{k} m_{k} (f) t^{k}$ (Morse numbers and the Morse polynomial); the handle chain complex of an index-ordered presentation has $C_{k}$ with basis the core classes and $H_{k} (C_{∙}) ≅ H_{k} (T^{2}; F)$ (The handle chain complex computes singular homology).

[F3]

For the surface endpoints $k = 0$ and $k = n - 1 = 1$ , the handle boundary coefficients in the core bases are given by the endpoint clause of the boundary-coefficient lemma: the coefficient of $\partial_{k + 1} e_{i}$ at $f_{j}$ is the intersection number of the attaching sphere of the upper handle with the belt sphere of the lower handle in the middle level (Handle boundary coefficients are attaching-belt intersection numbers). The matrix definition Attaching-belt intersection matrix of adjacent-index handles requires $1 \leq k \leq n - 2$ and has no surface case.

[F4]

For every field there is a unique $Q \in Z [t]$ with nonnegative coefficients and $M_{f} = P_{T^{2}, F} + (1 + t) Q$ ; $f$ is $F$ -perfect exactly when $m_{k} (f) = b_{k} (T^{2}; F)$ for all $k$ ; and the Euler characteristic identity $\sum_{k} (- 1)^{k} m_{k} (f) = χ (T^{2})$ holds (Morse polynomial identity, Perfect Morse function over a field, Morse Euler characteristic identity, Poincare polynomial of a space and of a pair over a field).

Verification

technique · direct-geometric-computation

1.1F1F2given

On the torus the gradient of $f$ is $- 2 π (\sin 2 π x, \sin 2 π y)$ , which vanishes exactly when both coordinates lie in ${0, \frac{1}{2}}$ ; at such a point the Hessian is $diag (- 4 π^{2} \cos 2 π x, - 4 π^{2} \cos 2 π y)$ , whose diagonal entries are both negative at $(0, 0)$ , of opposite signs at the two mixed points, and both positive at $(\frac{1}{2}, \frac{1}{2})$ . Hence the critical points are these four points, all nondegenerate, with indices $2, 1, 1, 0$ , and by [F2], $M_{f} (t) = 1 + 2 t + t^{2} .$

2.1F2step 1.1

By [F2] the index-ordered presentation of $f$ has $m_{0} = 1$ handle of index $0$ , $m_{1} = 2$ of index $1$ and $m_{2} = 1$ of index $2$ ; its handle chain complex over $F$ has $\dim_{F} C_{0} = 1$ , $\dim_{F} C_{1} = 2$ , $\dim_{F} C_{2} = 1$ .

3.1F3step 2.1

First boundary: the $1$ -handles are attached to the single $0$ -handle along two feet on its boundary circle $S^{1}$ , and the attaching $0$ -sphere $S^{0}$ is oriented as the boundary of the attaching interval, so the two feet contribute with opposite signs and their intersection numbers with the belt circle cancel; by the endpoint coefficient clause of [F3] this gives zero coefficients, so $\partial_{1} = 0$ . Equivalently, each $1$ -handle is a band gluing the two marked points with opposite orientations, and the two feet lie in the same component of the connected boundary.

3.2F3step 2.1algebra

Second boundary: just below the maximum $(0, 0)$ , the sublevel is the torus with an open disk removed. Each $1$ -handle is an untwisted band in this oriented surface. Its outgoing sides $D^{1} \times S^{0}$ are both in the boundary, and its belt sphere consists of their two midpoints. The remaining $2$ -handle caps this boundary circle, which traverses the two sides of each band in opposite core directions: this follows from the boundary orientation of the rectangle $D^{1} \times D^{1}$ . With the belt-point orientations compatible with the oriented core, the two local intersection signs are opposite. Thus every coefficient of $\partial_{2}$ is zero by [F3], integrally and over every field; modulo two the two points likewise cancel.

4.1F2F3F4step 3.1step 3.2

Since $\partial_{1} = \partial_{2} = 0$ , the handle chain complex equals its homology: $H_{0} ≅ F$ , $H_{1} ≅ F^{2}$ , $H_{2} ≅ F$ ; by [F2] the same holds for $H_{*} (T^{2}; F)$ , so $P_{T^{2}, F} (t) = 1 + 2 t + t^{2} = M_{f} (t) .$ Comparing in the Morse polynomial identity [F4], the correction polynomial is $Q = 0$ and $m_{k} (f) = b_{k} (T^{2}; F)$ for every $k$ : the function $f$ is $F$ -perfect, for every field $F$ .

5.1F4step 1.1step 4.1∎

Euler check: $\sum_{k} (- 1)^{k} m_{k} (f) = 1 - 2 + 1 = 0$ , and by the Euler identity of [F4] this equals $χ (T^{2})$ ; the same alternating sum of the Betti numbers $1 - 2 + 1$ is zero.

Remarks

The first interesting case. The surface has nontrivial $1$ -handles, while both endpoint boundary maps vanish for the standard perfect function. These endpoint computations use the boundary-coefficient lemma; the middle-index attaching-belt matrix definition has no surface case.
Coefficient independence. Because all boundary maps vanish integrally (the cancellation is by opposite signs, not merely mod two), the computation holds over every field at once; this contrasts with real projective space, where the torsion makes the answer depend on the characteristic.

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Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology, Part I Chapter 4 Section 4.4, printed pp. 88-91 (PDF pp. 98-100) (standard reference, not scraped)
Alexander Ritter, Morse Homology (Cambridge Part III lecture notes), Lecture 21, PDF pp. 96-101 (standard reference, not scraped)
C. T. C. Wall, Differential Topology, Sections 5.1-5.4, printed pp. 129-148 (PDF pp. 137-151) (standard reference, not scraped)