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Finiteness and additivity of the Euler characteristic

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a compact smooth $n$ -manifold, possibly with boundary.

(i) Each $\dim_{Q} H_{i} (M; Q)$ is finite and $H_{i} (M; Q) = 0$ for $i > n$ , so $χ (M)$ of Euler characteristic of a compact manifold is a well-defined integer.

(ii) If $A \subseteq M$ is a compact smooth submanifold, possibly with boundary, and $(M, A)$ is homotopy equivalent as a pair to a finite relative CW pair with $c_{k}$ relative $k$ -cells, then $χ (M) = χ (A) + \sum_{k} (- 1)^{k} c_{k} = χ (A) + \sum_{k} (- 1)^{k} \dim_{Q} H_{k} (M, A; Q) .$

(iii) If $M = M_{1} \cup_{N} M_{2}$ with $M_{1}, M_{2}$ compact smooth submanifolds, possibly with boundary, $N = M_{1} \cap M_{2}$ a common compact smooth submanifold, and the inclusions $N ↪ M_{i}$ cofibrations, then $χ (M) = χ (M_{1}) + χ (M_{2}) - χ (N) .$

Facts & Assumptions

Given: The Axiom of Choice and a compact smooth $n$ -manifold $M$ , possibly with boundary.

[F1]

The double $D M$ is a closed smooth $n$ -manifold. Its continuous folding map $q ([x, +]) = q ([x, -]) = x$ is well defined on the quotient and is a retraction onto the first labelled copy, with $q \circ i = {id}_{M}$ (The double of a smooth manifold with boundary, The double has a well-defined smooth structure).

[F2]

For a closed smooth manifold with an excellent Morse function, the handle chain complex of The handle chain complex computes singular homology is a complex of finite-dimensional $Q$ -vector spaces with exactly $m_{k} (f)$ generators in degree $k$ whose homology is $H_{k} (M; Q)$ ; hence those homology spaces are finite-dimensional and vanish above the dimension (Every compact smooth manifold admits an excellent Morse function, The singular chain complex and singular homology).

[F3]

Homotopy equivalences induce isomorphisms on singular homology with any coefficients, and the long exact sequence of a pair and the Mayer-Vietoris sequence are long exact sequences of $Q$ -vector spaces (Homotopy equivalences induce isomorphisms on singular homology, Long exact sequence of a pair, Mayer–Vietoris sequence in singular homology).

[F4]

Alternating-dimension lemma: for a long exact sequence $\dots \to A_{k} \to B_{k} \to C_{k} \to A_{k - 1} \to \dots$ of finite-dimensional vector spaces vanishing outside a finite range, one has $\sum_{k} (- 1)^{k} (\dim A_{k} - \dim B_{k} + \dim C_{k}) = 0$ , and the intermediate spaces are finite-dimensional with the same vanishing range (Rank bookkeeping for a long exact sequence of finite-dimensional vector spaces, 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).

[F5]

For a CW pair, relative cellular chains have one generator per relative cell and compute relative singular homology (Relative cellular homology computes relative singular homology). For a finite chain complex, writing $Z_{k} = \ker d_{k}$ and $B_{k} = im d_{k + 1}$ gives $\dim C_{k} = \dim H_{k} + \dim B_{k} + \dim B_{k - 1}$ ; alternating summation cancels the boundary dimensions. Thus the alternating relative cell count equals the alternating rational relative Betti sum, even when the base subcomplex itself has infinitely many cells.

[F6]

The smooth spaces $M_{i}, N$ in (iii) are CGWH under the assumed AC (Smooth manifolds have CW homotopy type), so the cofibration interface applies. For a closed cofibration $A ↪ X$ , the homotopy extension property supplies a retraction $X \times I \to X \times {0} \cup A \times I$ (Cofibrations are characterized by a retraction of the mapping cylinder strip).

Proof

1.1F1F2givenalgebra

For (i), the empty case is immediate. Otherwise the folding retraction [F1] gives $q_{*} i_{*} = id$ on rational homology, so $H_{i} (M; Q)$ embeds as a direct summand of $H_{i} (D M; Q)$ . Under the assumed Axiom of Choice, choose an excellent Morse function on the closed double and apply [F2]. Its handle complex is finite-dimensional and concentrated in degrees $0, \dots, n$ ; its homology, and hence the summand for $M$ , is finite-dimensional and zero above $n$ . This establishes well-definedness without presupposing $χ (M)$ .

1.2F6construct

For (iii), replace the glued space by the double mapping cylinder $P = M_{1} \cup_{N \times {0}} (N \times I) \cup_{N \times {1}} M_{2}$ . We verify that collapsing the cylinder is a homotopy equivalence $P \to M$ . Let $C = M_{1} \cup_{N} (N \times I)$ , with collapse $f : C \to M_{1}$ . By [F6] extend the track $H (n, t) = [n, t]$ and the initial inclusion of $M_{1}$ to $H : M_{1} \times I \to C$ . Set $j = H_{1}$ , so $j (n) = [n, 1]$ . Then $f H$ joins ${id}_{M_{1}}$ to $f j$ relative to $N$ . On $C$ , use $H (x, t)$ for $x \in M_{1}$ and $[n, s + t (1 - s)]$ for $[n, s]$ in the cylinder; the formulas agree at $s = 0$ , define a homotopy from ${id}_{C}$ to $j f$ , and fix the free end $N \times {1}$ . Gluing these maps and homotopies to the identity of $M_{2}$ proves the asserted equivalence. The compact subspaces $M_{i}$ are closed in $M$ , so their pushout topology is the topology of $M_{1} \cup M_{2} = M$ by finite closed pasting.

2.1F3F4F5step 1.1algebra

For (ii), step 1.1 applies to both $M$ and $A$ . In the pair long exact sequence [F3], $H_{k} (M, A; Q)$ lies between a quotient of $H_{k} (M; Q)$ and a subspace of $H_{k - 1} (A; Q)$ ; it is therefore finite-dimensional and vanishes outside a finite range. Alternating summation of this exact sequence gives $χ (M) - χ (A) = \sum_{k} (- 1)^{k} \dim H_{k} (M, A; Q)$ by [F4], applied after cyclically relabelling the three terms if necessary. The given equivalence of pairs induces isomorphisms on these relative groups: the absolute homology maps are isomorphisms by [F3], and exactness of the pair sequences gives injectivity and surjectivity of the relative maps by lifting and subtracting successive neighbouring classes. Now [F5] computes the relative alternating sum as $\sum_{k} (- 1)^{k} c_{k}$ , proving both equalities without requiring a finite absolute CW structure on $A$ .

3.1F3F4step 1.1step 1.2algebra∎

The open subsets $U = M_{1} \cup (N \times [0, 2 / 3))$ and $V = M_{2} \cup (N \times (1 / 3, 1])$ cover $P$ . They deformation retract to $M_{1}, M_{2}$ , while $U \cap V = N \times (1 / 3, 2 / 3)$ retracts to $N$ . Thus [F3] and step 1.2 give a Mayer–Vietoris sequence with homology terms those of $N$ , $M_{1} ⊔ M_{2}$ and $M$ . All terms are finite-dimensional and vanish outside a finite range by step 1.1. Applying [F4] at $t = - 1$ gives $χ (N) - χ (M_{1}) - χ (M_{2}) + χ (M) = 0$ , as required.

Depends on

Used by

Cited to discharge well-definedness by Euler characteristic of a compact manifold.

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Sources

John W. Milnor, Topology from the Differentiable Viewpoint (complete 76-page PDF, including the appendix Classifying 1-manifolds) (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Section 2.2 (standard reference, not scraped)
Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete PDF) (standard reference, not scraped)