Euler characteristic of a compact manifold

Let $M$ be a compact smooth $n$-manifold, possibly with boundary (Smooth manifolds and their smooth charts in the boundaryless case, Smooth charts, atlases, and structures with boundary in the boundary case). Its Euler characteristic is $\chi (M):={\sum }_{i=0}^n(-1{)}^i{\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})\in \mathbb{Z},$ the alternating sum of the rational Betti numbers of the singular homology of $M$ (The singular chain complex and singular homology, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, The rationals as equivalence classes of pairs of integers). The empty manifold has $\chi (\mathrm{\varnothing })=0$, the empty sum. The sum is finite because ${\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})<\mathrm{\infty }$ for every $i$ and $H_i(M;\mathbb{Q})=0$ for $i>n$; that finiteness is proved on this page in Finiteness and additivity of the Euler characteristic ↗, which is why it is recorded as the well-definedness pointer rather than assumed here. The definition itself uses no orientation and no choice of field or coefficient system.

When $M$ has a finite CW model, $\chi (M)$ agrees with its cell-count Euler characteristic (Euler characteristic of a finite CW complex): the finite rational cellular complex has one generator per cell, and alternating rank-nullity cancels boundary dimensions. Cellular homology (Cellular homology computes singular homology) therefore identifies that cell count with the rational Betti alternating sum. The integral Euler-Poincare formula (Euler–Poincare formula for finite CW complexes) gives the same count. The relative version is proved in Finiteness and additivity of the Euler characteristic ↗. The cited well-definedness proposition assumes the Axiom of Choice (The Axiom of Choice) to obtain an excellent Morse function. The formula for $\chi$ makes no selection of auxiliary data.