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Weak Morse inequalities

Statement

Assume ${A C}_{ω}$ . In the situation of the Morse polynomial identity (Morse polynomial identity), $m_{k} (f) \geq b_{k} (M; F)$ for every $k$ ; that is, the number of critical points of index $k$ is at least the $k$ -th Betti number over $F$ .

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , a Morse function $f : M \to R$ , a field $F$ , and the correction polynomial $Q (t) = \sum_{k} q_{k} t^{k}$ with $q_{k} \geq 0$ of the Morse polynomial identity.

[F1]

$M_{f} (t) = P_{M, F} (t) + (1 + t) Q (t)$ with $Q \in Z [t]$ of nonnegative coefficients (Morse polynomial identity), and the Morse and Betti numbers are the coefficients of $M_{f}$ and $P_{M, F}$ (Morse numbers and the Morse polynomial, Poincare polynomial of a space and of a pair over a field).

Proof

technique · coefficient-comparison

1.1F1givenalgebra

Comparing the coefficient of $t^{k}$ in $M_{f} - P_{M, F} = (1 + t) Q$ gives $m_{k} - b_{k} = q_{k} + q_{k - 1}$ with $q_{- 1} : = 0$ .

2.1step 1.1algebra∎

Since $q_{k} \geq 0$ and $q_{k - 1} \geq 0$ , step 1.1 gives $m_{k} - b_{k} \geq 0$ , that is $m_{k} \geq b_{k}$ , for every $k$ .

Remarks

The weak inequalities follow from $m_{k} - b_{k} = q_{k} + q_{k - 1}$ ; the strong inequalities retain the separate condition $q_{k} \geq 0$ .

Depends on

Used by

Total critical point lower bound Corollary
Real projective space shows coefficient-dependent perfectness Example

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)
Alexander Ritter, Morse Homology (Cambridge Part III lecture notes), Lecture 21, PDF pp. 96-101 (standard reference, not scraped)