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Morse Euler characteristic identity

Statement

Assume ${A C}_{ω}$ . Let $M$ be a closed smooth $n$ -manifold and $f : M \to R$ a Morse function. Then $\sum_{p \in Crit (f)} (- 1)^{ind (p)} = χ (M),$ where $χ (M)$ is the Euler characteristic computed from a finite CW model homotopy equivalent to $M$ (Euler characteristic of a finite CW complex); the Euler-Poincare formula makes it independent of the chosen structure and equal to the alternating sum of the Betti numbers (Euler–Poincare formula for finite CW complexes). The identity is independent of the coefficient field, and it holds for every closed smooth manifold, orientable or not.

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , a Morse function $f : M \to R$ , the Morse polynomial $M_{f} (t) = \sum_{k} m_{k} (f) t^{k}$ and the alternating critical-point sum $\sum_{k} (- 1)^{k} m_{k} (f) = M_{f} (- 1)$ .

[F1]

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

[F2]

$Q$ is a field (The rationals form a field, The rationals as equivalence classes of pairs of integers).

[L1]

A finite CW complex has Euler characteristic equal to its alternating cell count, which by the Euler-Poincare formula equals $\sum_{n} (- 1)^{n} rank H_{n} (X; Z)$ (Euler characteristic of a finite CW complex, Euler–Poincare formula for finite CW complexes); applied to the finite CW model of $M$ furnished by its handle presentation (A handle decomposition gives a relative CW complex), the rational equality is established by the finite chain calculation in step 1.2 below.

[F3]

Evaluation at a ring element is additive and multiplicative: $(P + Q) (a) = P (a) + Q (a)$ and $(P Q) (a) = P (a) Q (a)$ (Evaluation and roots of a polynomial in a commutative target ring).

[F4]

Critical values can be separated without changing critical points or Hessians (For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians). An excellent Morse function on the compact collared triad determines a finite handle presentation with exactly one handle of index $ind (p)$ per critical point, and for the closed case this presentation is obtained by the empty-face convention (Morse functions and handle decompositions correspond).

[F5]

A handle decomposition of a compact manifold gives a relative CW model homotopy equivalent to the manifold, with one cell per handle, of the same index (A handle decomposition gives a relative CW complex).

[F6]

For a finite CW complex, $χ (X) = \sum_{n} (- 1)^{n} rank H_{n} (X; Z)$ , and the cellular chain complex computes singular homology, the rational Betti alternating sum follows by cancellation of boundary ranks in its finite rational cellular complex (Euler–Poincare formula for finite CW complexes, Euler characteristic of a finite CW complex, Cellular homology computes singular homology, Cellular homology). Homotopy equivalences induce homology isomorphisms over every coefficient group (Homotopy equivalences induce isomorphisms on singular homology).

Proof

technique · evaluation-at-minus-one

1.1F1F2given

By [F2] we may apply [F1] with $F = Q$ : there is $Q \in Z [t]$ with nonnegative coefficients and $M_{f} (t) = P_{M, Q} (t) + (1 + t) Q (t)$ .

1.2F5F6L1algebra

The handle chain complex computes $H_{*} (M; Q)$ and has dimensions $m_{k} (f)$ (The handle chain complex computes singular homology). An index-ordered presentation gives a finite CW model by [F5], so its cell-count Euler characteristic is $\sum_{k} (- 1)^{k} m_{k} (f)$ . For any finite rational chain complex, choose a basis of each boundary space, extend it to a basis of the cycle space, and lift a basis of the next boundary space to the chain group. This gives $\dim C_{k} = \dim H_{k} + \dim B_{k} + \dim B_{k - 1}$ ; taking the alternating sum cancels both boundary terms. Applied to the model cellular complex, it gives $χ (M) = \sum_{k} (- 1)^{k} \dim_{Q} H_{k} (M; Q)$ , using [F6]. Homotopy invariance makes this independent of the finite model.

2.1F3L1step 1.1

Evaluating the identity of step 1.1 at $t = - 1$ and using [F3] gives $M_{f} (- 1) = P_{M, Q} (- 1) + (1 + (- 1)) Q (- 1) = P_{M, Q} (- 1)$ , that is $\sum_{p \in Crit (f)} (- 1)^{ind (p)} = \sum_{k} (- 1)^{k} \dim_{Q} H_{k} (M; Q) = χ (M),$ the last equality by [L1].

3.1F1step 2.1

Field independence: repeating steps 1.1–2.1 with an arbitrary field $F$ in place of $Q$ gives $\sum_{k} (- 1)^{k} b_{k} (M; F) = M_{f} (- 1) = χ (M)$ , so the alternating sum of the Betti numbers is the same for every field; in particular the identity does not depend on $F$ .

4.1F4F5F6step 1.2step 2.1∎

For the handle-side count, rescale $f$ into $(0, 1)$ when $M \neq \emptyset$ , separate its critical values by [F4], and use the index-ordered handle presentation already constructed in step 1.2. Its finite CW model has $m_{k} (f)$ cells of dimension $k$ , so its Euler characteristic is $M_{f} (- 1)$ . The empty manifold gives the empty model and zero on both sides. This confirms the identity without asserting that the cell model is a CW structure on the original manifold itself.

Remarks

Independence of the field. Both the Morse numbers (geometric) and $χ (M)$ are field independent, and step 3.1 shows the intermediate Betti alternating sums are too; this is why the Euler identity survives while the weak and strong inequalities fail to be field independent.
The role of orientability. Neither the handle presentation nor the cell-count computation uses an orientation; the identity therefore holds for nonorientable closed manifolds as well, and the mod-two handle chain complex would compute the same alternating count.

Depends on

Used by

The Euler number of the tangent bundle is the Euler characteristic Corollary
Euler equality alone does not imply perfectness Counterexample
A Morse function on the torus is perfect over every field Example
Real projective space shows coefficient-dependent perfectness Example
Morse inequalities and perfectness depend on the coefficient field Remark
Poincare-Hopf for closed manifolds Theorem

Dependency tree · two levels

82 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)
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)
C. T. C. Wall, Differential Topology, Sections 5.1-5.4, printed pp. 129-148 (PDF pp. 137-151) (standard reference, not scraped)