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Perfectness, vanishing correction, and vanishing handle boundaries

Statement

Assume ${A C}_{ω}$ . Let $M$ be a closed smooth $n$ -manifold, let $f : M \to R$ be Morse, let $F$ be a field, let $Q$ be the correction polynomial of the Morse polynomial identity and let $(C_{∙}, \partial_{∙})$ be a handle chain complex of $(M, f, F)$ . The following are equivalent:

(i) $f$ is $F$ -perfect;

(ii) $Q = 0$ ;

(iii) every handle boundary map vanishes, $\partial_{k} = 0$ for all $k$ (equivalently $\dim_{F} im \partial_{k + 1} = 0$ for all $k$ ).

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , a Morse function $f$ , a field $F$ , the unique correction polynomial $Q (t) = \sum_{k} q_{k} t^{k}$ with $q_{k} \geq 0$ of the identity $M_{f} = P_{M, F} + (1 + t) Q$ , and a handle chain complex $(C_{∙}, \partial_{∙})$ .

[F1]

$M_{f} = P_{M, F} + (1 + t) Q$ with $Q \in Z [t]$ having nonnegative coefficients, and $Q$ is the unique such polynomial (Morse polynomial identity).

[F2]

$f$ is $F$ -perfect exactly when $m_{k} (f) = b_{k} (M; F)$ for all $k$ , equivalently when $M_{f} = P_{M, F}$ (Perfect Morse function over a field).

[F3]

The handle chain complex has $\dim_{F} C_{k} = m_{k} (f)$ , it is a chain complex with $\partial_{k - 1} \partial_{k} = 0$ , and $H_{k} (C_{∙}) ≅ H_{k} (M; F)$ , so $\dim_{F} H_{k} (C_{∙}) = b_{k} (M; F)$ ; all these spaces are finite-dimensional (The handle chain complex computes singular homology).

[L1]

For a linear map $T : V \to W$ with $V$ finite-dimensional, $\dim_{F} V = \dim_{F} \ker T + \dim_{F} im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ , Rank and nullity of a linear map with finite-dimensional domain).

[L2]

The rank bookkeeping for a short exact sequence $0 \to A \to B \to C \to 0$ of finite-dimensional $F$ -vector spaces gives $\dim_{F} B = \dim_{F} A + \dim_{F} C$ : it is the one-term case of the alternating partial-sum identity with all other terms zero and injective first map (Rank bookkeeping for a long exact sequence of finite-dimensional vector spaces).

[L3]

A subspace of a finite-dimensional space is finite-dimensional (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

Proof

technique · rank-comparison

1.1F1F2given

(i) $\Leftrightarrow$ (ii): if $f$ is $F$ -perfect, then $M_{f} = P_{M, F}$ by [F2], so $(1 + t) Q = 0$ and uniqueness in [F1] gives $Q = 0$ (the zero polynomial is a solution). Conversely, if $Q = 0$ , then $M_{f} = P_{M, F}$ by [F1], and coefficient comparison gives $m_{k} (f) = b_{k} (M; F)$ for every $k$ , which is perfectness by [F2].

1.2F3F2given

(iii) $\Rightarrow$ (i): if $\partial_{k} = 0$ for every $k$ , then $H_{k} (C_{∙}) = C_{k}$ and hence, by [F3], $m_{k} (f) = \dim_{F} C_{k} = \dim_{F} H_{k} (C_{∙}) = b_{k} (M; F)$ for every $k$ ; by [F2] the function $f$ is $F$ -perfect.

1.3F3L1L2L3given

(i) $\Rightarrow$ (iii): for every $k$ , [L1] applied to $\partial_{k} : C_{k} \to C_{k - 1}$ gives $\dim_{F} C_{k} = \dim_{F} \ker \partial_{k} + \dim_{F} im \partial_{k},$ and the same rank-nullity identity applied to the short exact sequence $0 \to im \partial_{k + 1} \to \ker \partial_{k} \to H_{k} (C_{∙}) \to 0$ (whose terms are finite-dimensional by [F3] and [L3]) gives, by [L2], $\dim_{F} \ker \partial_{k} = \dim_{F} im \partial_{k + 1} + \dim_{F} H_{k} (C_{∙}) .$ Substituting and using [F3] yields $m_{k} (f) - b_{k} (M; F) = \dim_{F} im \partial_{k + 1} + \dim_{F} im \partial_{k} \geq 0.$ If $f$ is $F$ -perfect, the left side vanishes for every $k$ ; both terms on the right are nonnegative dimensions, so both vanish for every $k$ , that is $im \partial_{k + 1} = 0$ and $im \partial_{k} = 0$ for all $k$ , which is (iii).

2.1step 1.1step 1.2step 1.3∎

Steps 1.1, 1.2 and 1.3 give (i) $\Leftrightarrow$ (ii) and (i) $\Leftrightarrow$ (iii), so all three statements are equivalent. In particular perfectness over $F$ is exactly the vanishing of all handle boundaries over $F$ ; over the integers the corresponding boundary maps need not vanish, since a nonzero integral boundary coefficient can vanish after reduction modulo the characteristic of $F$ .

Remarks

Measure of the loss. The identity of step 1.3 exhibits the defect $m_{k} - b_{k}$ as the total dimension of the two boundary maps meeting in degree $k$ ; this is the handle-side reading of the correction polynomial.
No identification with the Morse differential. The vanishing is asserted for the handle boundary maps of the handle chain complex constructed on this page; the trajectory-count definition of the Morse differential and its comparison with these maps belong to the later Morse-homology page and are not used here.

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Sources

Liviu Nicolaescu, An Invitation to Morse Theory (2nd ed.), Chapter 2 Section 2.3, printed pp. 46-53 (PDF pp. 56-63) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Chapter 12 Section 5, printed pp. 489-493 (PDF pp. 501-505) (standard reference, not scraped)