nLab Euler characteristic

Idea

The Euler characteristic of an object is – if it exists – its categorical dimension.

Definitions

Of a chain complex

There are two definitions of the Euler characteristic of a chain complex. Let $R$ be a commutative ring and let $V$ be a chain complex of $R$-modules.

Definition

If each $V_n$ is finitely generated and projective, then the Euler characteristic of $V$ is the alternating sum of their ranks, if this is finite:

$$\chi(V) \,\coloneqq\, \sum_{n \in \mathbb{Z}} (-1)^n \, rk_R V_n \,.$$

Definition

If the homology of $V$ consists of finitely generated projective modules, then the Euler characteristic of $V$ is the alternating sum of its Betti numbers (the ranks of its homology modules), if this is finite:

$$\chi(V) \,\coloneqq\, \sum_{n \in \mathbb{Z}} (-1)^n \, rk_R H_n(V) \,.$$

When both of these are defined, they are equal. This is a consequence of the functoriality of the categorical definition (Definition ).

Remark

Definition shows that the Euler characteristic of chain complexes is invariant under the natural notion of equivalence of chain complexes: (zig-zags of) quasi-isomorphisms.

Of a topological space (or $\infty$-groupoid)

Definition

For $X$ a topological space and $R$ some ring, its Euler characteristic over $R$ is the Euler characteristic, according to Def. , of its homology chain complex (for instance singular homology), if this is finite: the alternating sum of its Betti numbers

(1)$$\begin{aligned} \chi(X) & \;\coloneqq\; \sum_{n \in \mathbb{N}} \, (-1)^n \, rk_{\mathbb{Z}} H_n(X, \mathbb{Z}) \\ & \;=\; \sum_{n \in \mathbb{N}} \, (-1)^n \, dim_{\mathbb{Q}} H_n(X, \mathbb{Q}) \end{aligned} \,.$$

This is shown for $R = \mathbb{Z}$ the integers (integral cohomology), and equivalently for $R = \mathbb{Q}$ rational numbers (rational cohomology).
(Choosing a ring with torsion, however, might result in a different “Euler characteristic”.)

This definition is usually known as the Euler-Poincaré formula.

Remark

This is the special case of the formula for the Lefschetz number: The Euler characteristic (1) is the Lefschetz number of the identity map, see there.

Historically earlier was:

Definition/Proposition

Let $X$ be a finite CW-complex. Write $cell(X)_k$ for the set of its $k$-cells. Then the Euler characteristic of $X$ is

$$\chi(X) \;=\; \sum_{k \in \mathbb{N}} (-1)^k \vert cell(X)_k \vert \,.$$

This is equivalently the Euler characteristic of the cellular chain complex of $X$ according to Definition . Thus, since the homology of $X$ can be computed with cellular homology, this Euler characteristic agrees with the previous one.

In the special case that $X$ is a surface, Def. reduces to the historical definition by Leonhard Euler, which implicitly was known already to Descartes around 1620:

Definition/Proposition

Let $X$ be a convex polyhedron. Then its Euler characteristic is

$$\chi(X) \;\coloneqq\; \vert Vertices(X)\vert - \vert Edges(X)\vert + \vert Faces(X)\vert \;\in\; \mathbb{Z} \,,$$

hence the number of vertices minus the number of edges plus the number of faces in the polyhedron.

In particular if $X$ may be embedded into the 2-sphere, this means that

$$2 = \vert Vertices(X)\vert - \vert Edges(X)\vert + \vert Faces(X)\vert \,.$$

By removing one point from the 2-sphere not contained in $X$, the result may be thought of as a planar graph. This has one face less than $X$ had (the one containing the point which was removed). Hence

Corollary

(Euler formula for planar graphs)

For a planar graph $\Gamma$ we have

$$1 = \vert Vertices(\Gamma)\vert - \vert Edges(\Gamma)\vert + \vert Faces(\Gamma)\vert \,.$$

Of an object in a symmetric monoidal category

All the definitions considered so far can be subsumed by the following general abstract one.

Definition

The Euler characteristic of a dualizable object in a symmetric monoidal category is its categorical dimension: the trace of its identity morphism.

Explicitly, this means the Euler characteristic of an object $X$ is the composite

$$S \xrightarrow{\eta} X \otimes X^* \xrightarrow{\cong} X^* \otimes X \xrightarrow{\varepsilon} S$$

where $\eta$ and $\varepsilon$ are the unit and counit of the dual pair $(X,X^*)$, and $S$ is the unit object of the symmetric monoidal category. This subsumes the previous definitions as follows:

• In the category of chain complexes over a ring $R$, an object is dualizable if it is finitely generated overall, and projective in each degree. Its dual is then given by $(X^*)_n = Hom_R(X_{-n},R)$. The unit $\eta$ picks out $\sum x \otimes x^*$, where $\{x\}$ encompasses a basis for each $X_n$ and $\{x^*\}$ is the dual basis.

  The symmetry isomorphism $X \otimes Y \xrightarrow{\cong} Y \otimes X$ introduces a sign $x\otimes y \mapsto (-1)^{|y|} y\otimes x$, so that when we then evaluate, we get a contribution of $1$ for each $x$ of even degree and $-1$ for each $x$ of odd degree. Thus we recover Def. . Note that the unit object is $R$ itself in degree zero, so that we see $\chi(X)$ only as an element of $R$ (so, for instance, the Euler characteristic in this sense of a rank-$p$ free $(\mathbb{Z}/p)$-module is zero.

• In the derived category of chain complexes over $R$, an object is dualizable if it is quasi-isomorphic to one of the form above. A similar argument shows that its Euler characteristic is then computed as in Def. .

The Euler characteristic of a topological space or $\infty$-groupoid can also be defined directly with this approach, without a detour into homology. Namely, if $X$ is any Euclidean Neighborhood Retract?, such as a finite CW complex or a smooth manifold, then its suspension spectrum $\Sigma_+^\infty X$ is dualizable in the stable homotopy category. Its dual is the Thom spectrum of its stable normal bundle, with the unit of the duality being the Thom collapse map. Its categorical Euler characteristic is then an endomorphism of the sphere spectrum, which can be identified with an integer (via $\pi_0^s(S) = \mathbb{Z}$). See around DoldPuppe, corollary 4.6).

Since the categorical definition is purely in terms of the symmetric monoidal structure, Euler characteristics are preserved by any symmetric monoidal functor (as long as enough of its structure maps are isomorphisms). Since chains and homology can be made into symmetric monoidal functors, it follows that all the ways of defining the Euler characteristic of a space agree.

See (PontoShulman) and the discussion at Thom spectrum for more on this.

Homotopy cardinality ($\infty$-groupoid cardinality)

The above Euler characteristic of a topological space is the alternating sum over sizes of homology groups. Similar in construction is the alternating product of sizes of homotopy groups. This goes by the name ∞-groupoid cardinality or homotopy cardinality . But below we shall see that Euler characteristic of higher categories interpolates between this homotopical and the above homological notion.

Definition

For $X$ a topological space / homotopy type / ∞-groupoid, its homotopy cardinality or ∞-groupoid cardinality is – if it exists – the rational number given by the alternating product of cardinalities of homotopy groups

$$\chi_{homotop}(X) \;\coloneqq\; \sum_{[x] \in \pi_0(X)} \prod_{k =1}^\infty (|\pi_k(X,x)|)^{(-1)^k} \,.$$

Of posets, groupoids and categories

The process of sending a category $C$ to its geometric realization of categories ${\vert C \vert} \in$ Top $\simeq$ ∞Grpd is a way to present topological spaces, and hence ∞-groupoids, by a category: we can think of $\vert C \vert$ as the Kan fibrant replacement of $C$: the universal solution to weakly inverting all morphisms of $C$.

Up to the relevant notion of equivalence in an (infinity,1)-category (which is weak homotopy equivalence), every ∞-groupoid arises as the nerve/geometric realization of a category. In fact one can assume the category to be a poset. (This follows from the existence of the Thomason model structure, as discussed in more detail there.)

Since the combinatorial data in a category and all the more in a poset $C$ is much smaller than in that of its Kan fibrant replacement $\vert C \vert$, it is of interest to ask if one can read off the Euler characteristic $\chi(\vert C \vert)$ already from $C$ itself. This is indeed the case:

Of finite categories

Definition

Let $C$ be a finite category. A weighting on $C$ is a function

$$k^{(-)} \colon ob C \to \mathbb{Q}$$

satsifying

$$\forall a \in ob C \;:\; \sum_{b \in ob C} \vert C(a,b) \vert k^b = 1 \,,$$

where $\vert C(a,b)\vert$ is the cardinality of the hom-set $C(a,b)$. A coweighting on $C$ is a weighting on the opposite category $C^{op}$.

If $C$ admits both a weighting $\{k^a\}$ and a coweighting $\{k_a\}$, then its Euler characteristic is

$$\chi(C) := \sum_{a \in ob C} k^a = \sum_{a \in ob C} k_a \;\;\; \in \mathbb{Q} \,.$$

The definition of Euler characteristic of posets appears for instance in (Rota). For groupoids it has been amplified in BaezDolan. The joint generalization to categories is due to (Leinster), where the above appears as def. 2.2.

Remark

Notice that this $\chi(C)$ is in general not an integer, but a rational number. However in sufficiently well-behaved cases, discussed below, $\chi(C)$ coincides with the topological Euler characteristic $\chi(\vert C \vert)$ of its geometric realization. Since that is integral, in these cases also $\chi(C)$ is.

Of enriched and higher categories

Let $V$ be an good enrichment category (for instance a cosmos) which itself comes equipped with a good notion of cardinality, in the form of a monoidal functor

$$|-| : V \to \mathbb{R} \,.$$

Then the above formula for the Euler characteristic of a category verbatim generalizes to $V$-enriched categories. The ordinary case is recovered for $V =$ FinSet and $|-| : FinSet \to \mathbb{R}$ the ordinary cardinality operation.

Since strict infinity-categories can be understood as arising from iterative enrichment

$$Str2Cat \simeq Cat-Cat$$

$$Str3Cat \simeq Str2Cat-Cat$$

etc, this gives a notion of Euler characteristic of strict $\infty$-categories, hence in particular of strict infinity-groupoids.

One should be able to show that applied to strict $\infty$-groupoids this does reproduce homotopy cardinality.

Properties

Compatibility with homotopy colimits

Euler characteristic behaves well with respect to the basic operations in homotopy theory.

Proposition

In a symmetric monoidal triangulated category with dualizable objects $X, Y, Z$, if

$$\array{ Z &\to& Y \\ \downarrow && \downarrow \\ X &\to& W }$$

is a homotopy pushout, then the tractial Euler characteristic of $W$ exists and is

$$\chi(W) = \chi(X) + \chi(Y) - \chi(Z) \,.$$

This is due to (May, 1991).

Relations between the various definitions

The following propositions assert that and how the various definitions of Euler characteristics all suitably agree when thez jointly apply.

Proposition

For $X$ a compact manifold let $P_T(X)$ be the poset of inclusions of simplices of a triangulation $T$ of $X$. Then the poset Euler characteristic of $P_T(X)$ coincides with the Euler characteristic of $X$ as a topological space

$$\chi(X) = \chi(P_T(X)) \,.$$

This appears as (Stanley, 3.8).

The following proposition asserts that the definition of Euler characteristic of a category is indeed consistent, in that it does compute the Euler characteristic, def. of the corresponding $\infty$-groupoid:

Proposition

Let $C$ be a finite poset or, slightly more generally, a finite skeletal category with no nontrivial endomorphisms.

Write $\vert C \vert \in$ Top $\simeq$ ∞Grpd for its geometric realization. Then

$$\chi(C) = \chi(\vert C \vert) \,.$$

For posets this is due to Philipp Hall, appearing as ([Stanley, prop. 3.8.5]). For finite categories this is (Leinster, cor. 1.5, prop. 2.11).

Proposition

For $X$ a manifold, regard its suspension spectrum

$$\Sigma^\infty X \in Ho(Spec)$$

as an object in the stable homotopy category. Then its Euler characteristic as an object of a symmetric monoidal category, def. coincides with its topological Euler characteristic, def. .

This is due to … (?)

Mike: Can the Euler characteristic of a category be recovered as the trace for a dualizable object in some symmetric monoidal category?

Relation between Euler characteristic and $\infty$-groupoid cardinality

For topological space / $\infty$-groupoids, there is both the notion of homological Euler characteristic as well as the notion of homotopy cardinality.

The latter looks a bit like the “exponential” of the former, so while similar to some extent they are very different notions, taken on face value. Still, the Euler characteristic of a category or rather that of a higher category does interpolate between the two notions:

Proposition

For $C$ a finite groupoid, its Euler characteristic as a category, def. , coincides with its homotopical Euler characteristic, def. or groupoid cardinality

$$\chi(C) = \chi_{homotop}(C) \,.$$

This is noted in (Leinster, example 2.7).

Example

For instance for $G$ a finite group let $B G \simeq K(G,1) \in$ Ho(Top) be its classifying space. This is the geometric realization both of the one-object groupoids $*//G$ as well as of some finite poset $C$.

$$B G \simeq |*//G| \simeq |C| \,.$$

By prop. we have that the categorical Euler characteristic of $C$ is the topological Euler characteristic of $B G$. But by prop. we have that the categorical Euler characteristic of $*//G$ is the homotopy cardinality of $B G$.

Typically for one and the same $\infty$-groupoid, Eucler characteristic and homotopy cardinality are never both well defined: if the series for one converges, that for the other does not.

However, by applying some standard apparent “tricks” on non-convergent series, often these can be made sense of after all, and then do agree with the other notion. For more on this see (Baez05).

Basic relations for Euler characteristics of topological spaces

Proposition

(Poincaré–Hopf theorem)
(…)

Proposition

(Euler characteristic of spheres)
The Euler characteristic of an even dimensional sphere is 2, that of an odd dimensional sphere vanishes:

$$\chi(S^{n}) \;=\; \left\{ \array{ 2 &\vert& n \, \text{is even} \\ 0 &\vert& \text{otherwise} } \right.$$

(e.g. Libgober, p. 16)

Proposition

(Euler characteristic of covering spaces)
If $X \xrightarrow{q} Y$ is a covering space over a connected topological space $Y$ of degree $deg(q) \in \mathbb{N}$ (and the ordinary homology of $X$ and $Y$ is finitely generated) then the Euler characteristic of $X$ is $deg(q)$ times that of $Y$:

$$\chi(X) \,=\, ord(G) \cdot \chi(Y) \,.$$

(e.g. topospaces here)

Proposition

(Euler characteristic of connected sums)
Let $X$ and $Y$ be closed manifolds (topological or differentiable) of even dimension. Then the Euler characteristic of their connected sum is the sum of their separate Euler characteristics, minus 2:

$$\chi \big( X \sharp Y \big) \;=\; \chi \big(X\big) + \chi \big(Y\big) -2$$

(e.g. here)

Gauss-Bonnet theorem

For $X$ an even-dimensional smooth manifold, its Euler characteristic may also be given by integration of infinitesimal data: this is the statement of the higher dimensional Gauss-Bonnet theorem.

Related concepts

• magnitude of an enriched category

• Euler class

• orbifold Euler characteristic

• arithmetic genus

• C-field tadpole cancellation

References

Review includes

• Jonathan Libgober, The Euler characteristic, Poincaré–Hopf theorem, and applications [pdf, pdf]

Textbook account:

• Edwin Spanier, p. 156 onwards in: Algebraic topology, McGraw-Hill (1966)

Efremovich and Rudyak shown that the Euler characteristic is (up to the overall multiplicative factor) the only additive homotopy invariant of a finite CW complexes:

• V. A. Efremovich, Yu. B. Rudyak, On the concept of the Euler characteristic, Uspehi Mat. Nauk 31:5(191) (1976), 239–240 MR458412, Russian pdf

The description of Euler characteristics are categorical traces in symmetric monoidal categories is discussed in section 4 of

• Albrecht Dold, Dieter Puppe, Duality, trace and transfer , Proceedings of the Steklov Institute of Mathematics, (1984), issue 4

Behaviour of tracial Euler characteristic under homotopy colimits is discussed in

• Peter May, The additivity of traces in triangulated categories K-theory (1991) (website)

Textbooks on combinatorial aspects of Euler characteristic include

• Richard P. Stanley, Enumerative Combinatorics , Vol. I, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, corrected reprint (1997)

• Gian-Carlo Rota, On the foundations of combinatorial theory I: theory of Möbius functions , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.

The Euler characteristic of a smooth manifold as its dimension in the stable homotopy category is discussed in example 3.7 of

• Kate Ponto, Mike Shulman, Traces in symmetric monoidal categories (pdf, slides).

See Thom spectrum for more on this

The Euler characteristic of groupoids – groupoid cardinality – has been amplified in

• John Baez, James Dolan, From Finite Sets to Feynman Diagrams (arXiv)

An exposition with an eye towards the relation between Euler characteristic and $\infty$-groupoid cardinality is in

• John Baez, The mysteries of counting: Euler characteristic versus homotopy cardinality (web) Lecture notes (2005)

A treatment of this relation using Morava K-theories is in

• Lior Yanovski, Homotopy Cardinality via Extrapolation of Morava-Euler Characteristics (arXiv:2303.02603)

The role of homotopy cardinality in quantization is touched on towards the end of

• Daniel Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups

The generalization of the definition of Euler characteristic from posets to categories is due to

• Tom Leinster, The Euler characteristic of a category (arXiv:0610260)

• Tom Leinster, The Euler characteristic of a category as a sum of a divergent series (arXiv:0707.0835)

The compatibility of Euler characteristic of categories with homotopy colimits is discussed in

• Thomas M. Fiore, Wolfgang Lück, Roman Sauer, Euler Characteristics of Categories and Homotopy Colimits [arXiv:1007.3868]

More on Euler characteristics of categories is in

• Kazunori Noguchi, The Euler characteristic of infinite acyclic categories with filtrations, arxiv/1004.2547

On “Negative sets” and Euler characteristic:

• André Joyal, Regle des signes en algebre combinatoire , Comptes Rendus Mathematiques de l’Academie des Sciences, La Societe Royale du Canada VII (1985), 285-290.

• Steve Schanuel, Negative sets have Euler characteristic and dimension , Lecture Notes in Mathematics 1488, Springer Verlag, Berlin, 1991, pp. 379-385.

• Daniel Loeb, Sets with a negative number of elements , Adv. Math. 91 (1992), 64-74.

Resumming divergent Euler characteristics:

• William J. Floyd, Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic , Invent. Math. 88 (1987), 1-29.

• R. I. Grigorchuk, Growth functions, rewriting systems and Euler characteristic , Mat. Zametki 58 (1995), 653-668, 798.

• James Propp, Exponentiation and Euler measure , Algebra Universalis 49 (2003), 459-471. (arXiv:math.CO/0204009).

• James Propp, Euler measure as generalized cardinality . (arXiv).

Euler characteristics of tame spaces:

• Lou van den Dries, Tame Topology and O-Minimal Structures Cambridge U. Press, Cambridge, 1998. Chapter 4.2: Euler Characteristic.

Euler characteristics of groups:

• G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups , Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.

• Jean-Pierre Serre, Cohomologie des groups discretes , Ann. Math. Studies 70 (1971), 77-169.

• Kenneth Brown, Euler characteristics, in Cohomology of Groups , Graduate Texts in Mathematics 182, Springer, 1982, pp. 230-272.

• O. Y. Viro, Some integral calculus based on Euler characteristic, in Topology and Geometry – Rohlin Seminar, Springer Lec. Notes in Math. 1346, 127–138 (1988) doi

Complex cardinalities:

• Andreas Blass, Seven trees in one , Jour. Pure Appl. Alg. 103 (1995), 1-21. (web)

• Robbie Gates, On the generic solution to $P(X) = X$ in distributive categories , Jour. Pure Appl. Alg. 125 (1998), 191-212.

• Marcelo Fiore, Tom Leinster, An objective representation of the Gaussian integers , Jour. Symb. Comp. 37 (2004), 707-716. (arXiv)

• Marcelo Fiore, Tom Leinster, Objects of categories as complex numbers , Adv. Math. 190 (2005), 264-277 (arXiv:0212377)

• Marcelo Fiore, Isomorphisms of generic recursive polynomial types , to appear in 31st Symposium on Principles of Programming Languages (POPL04). (ps)

• C. T. C. Wall, Arithmetic invariants of subdivision of complexes, Canad. J. Math. 18(1966), 92-96, doi, pdf

$\mathbf{A}^1$-Euler characteristic

• Arcila-Maya, Niny; Bethea, Candace; Opie, Morgan; Wickelgren, Kirsten; Zakharevich, Inna, Compactly supported $\mathbb{A}^1$-Euler characteristic and the Hochschild complex, Topology and its Applications 316 (2022) 108108 doi