is a fundamental invariant of , and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups , but actually one can work on a more intuitive level characterized by the following slogan:
The Euler characteristic is a homotopy-invariant generalization of cardinality.
More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:
- Cardinality: .
- Homotopy invariance: If , then .
- Inclusion-exclusion: Suppose is the union of two subcomplexes whose intersection is a subcomplex of both and . Then .
Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.
First, a word about the type of spaces we’re dealing with. It is too restrictive to work with simplicial complexes, since the axiom that two simplices intersect in a face is quite strong; for example, two points joined by two edges is not a simplicial complex, although the above result applies just fine to it. The ideas we’re describing apply to a generalization of simplicial complexes in which two simplices are allowed to intersect in a union of faces. For practical purposes, we’re dealing with spaces that can be constructed from a finite number of simplices by gluing along faces (or which are homotopy equivalent to such things), and we’ll refer to the interior of an -simplex as an -cell. For example, a solid tetrahedron is a -simplex and consists of one -cell, four -cells, six -cells, and four -cells.
As a sanity check, the inclusion-exclusion axiom implies that , so in particular the Euler characteristic of a finite set of points is just its cardinality.
The interval is contractible, hence . Another way to see this is that the interval can be written as the union of two intervals whose intersection is a point, so , hence . Note that can be written as a collection of points joined by edges, and .
The circle can be written as the union of two intervals whose intersection consists of two points, hence . If you believe that the Euler characteristic exists, it follows that is not contractible. Note that can be written as a collection of points joined by edges, and .
Now let be a connected finite -complex (graph) with vertices and edges. Any such graph has a spanning tree, and contracting along that spanning tree gives a wedge sum of circles. This wedge sum in turn can be written as where consists of an interval taken from each circle, is the closure of the complement of , which is contractible, and consists of points. It follows that
Intuitively, any edge can potentially cancel out a point via a homotopy, so it makes sense that edges count for . However, note that we cannot extend the Euler characteristic to spaces like the open interval without losing homotopy invariance.
The sphere can be written as the union of two disks (the northern and southern hemispheres), which are contractible, whose intersection is a circle, hence . Note that the sphere can be obtained by gluing a single point to a -cell, so the -cell appears to be contributing .
Why might -cells do this? One intuition is that -cells act as “higher edges” between edges. Take the space obtained from two points with two edges between them by gluing a -cell in between the edges. This space is the disk, but by contracting the -cell we get a deformation to an interval: in other words, the -cell has been used to cancel out a -cell, so it counts for minus what a -cell counts for, or . In the lingo of higher category theory, if -cells are morphisms between -cells, then -cells are -morphisms between -cells.
The torus can be written as the quotient of a square obtained by identifying opposite edges in the same orientation. (Intuitively this corresponds to curling up the square once into a cylinder, then joining the two ends of the cylinder together.) We can therefore write it as where is a small disc cut out of the middle of the square, is , and deformation retracts onto its -skeleton (the complex obtained by only looking at the cells of dimension or lower). This turns out to be the wedge of two circles, whence and . Note that the torus has a CW-structure with , and .
Let denote the -holed torus. By pinching off a part containing one hole, we can write it as where is a torus minus a disc, , and is minus a disc. Inclusion-exclusion gives , and we know that satisfies , so it follows by inclusion-exclusion that . Together with the base case , we conclude by induction that
It turns out that we can obtain by identifying certain pairs of sides in a -gon. It follows that it has a CW-structure with , and as expected.
Having built some intuition, we are now ready to dive into the proof. Our current intuition is that -cells can at least potentially cancel out -cells up to homotopy, hence if Euler characteristic behaves additively on cells, an -cell should count for by induction. To carry out this intuition we will work by induction and use inclusion-exclusion to remove all of the top-dimensional cells. Thus let be an -dimensional finite complex with different -cells. We want to remove a small ball from each -cell; balls are contractible, so . The intersection of this ball and the closure of its complement is the sphere , and after removing a small ball we can retract each -cell onto the lower-dimensional cells it’s attached to, so it follows that, if denotes the -skeleton of , we have
so it remains to compute . Again, by looking at two hemispheres, we can write as where are both homeomorphic to and is the equator, which is homeomorphic to . Explicitly, if , then we can take . It follows that
and since it follows by induction that , which gives
and the result follows by induction.
So how do we prove that the Euler characteristic exists? Given the existing discussion, a standard proof proceeds by showing that the Euler characteristic is equal to the alternating sum
where the are, depending on your preferences, the singular or simplicial homology groups, which are homotopy-invariant. This is usually done with a mildly unenlightening computation, but given what we know we can cheat: the above alternating sum satisfies inclusion-exclusion because of the Mayer-Vietoris sequence (and the fact that the Euler characteristic of an exact sequence vanishes), and it obviously satisfies the cardinality axiom, so by homotopy invariance it must be the Euler characteristic.
Of course there is much more to say on this subject. The Euler characteristic figures in many fundamental theorems of topology and geometry, including the Poincare-Hopf theorem and the Gauss-Bonnet theorem. I hope to say more about the Euler characteristic in future posts.