LECTURE . GENERALIZED EULER CHARACTERISTICS AND K-THEORY
Our starting place is with the theory of Euler characteristics.
.. Definition. Suppose X a finite CW complex, with a given CW decomposition
∪
X=
Xn × Dn .
n≥0
e Euler characteristic of X is the alternating sum of the number of cells:
∑
χ(X) =
(−1)n #Xn .
n≥0
is number ostensibly depends upon the chosen cell decomposition, but one of the first stories we learn about
homology is that, in fact, it’s an invariant of the space itself ! To remind ourselves why, let’s consider an algebraic Euler
characteristic.
.. Definition. Suppose E = (E∗ , ∂) a bounded complex of finite dimensional rational vector spaces. e Euler
characteristic of C is the alternating sum of the dimensions of the vector spaces therein:
∑
χ(E) =
(−1)n dim En .
n∈Z
is terminology is reasonable: if one forms the rational cellular complex of a finite CW-complex, then the Euler
characteristic of this complex is the Euler characteristic of the CW complex. Algebraically, we find an alternative
description of this Euler characteristic.
.. Proposition (Poincaré–Hopf ). Suppose E = (E∗ , ∂) a bounded complex of finite dimensional rational vector
spaces with homology groups H∗ . e Euler characteristic of E is then given by the alternating sum of the dimensions of
the homology vector spaces:
∑
χ(E) =
(−1)n dim Hn .
n∈Z
Proof. Let’s assume that E in concentrated in degrees [a, b]. Since dim Hn = dim ker ∂n − dim im ∂n+1 , we have
b
b
∑
∑
n
χ(E) =
(−1) dim ker ∂n +
(−1)n+1 dim im ∂n+1
n=a
= (−1)a dim ker ∂a +
n=a
b
∑
(−1)n (dim ker ∂n + dim im ∂n ) + (−1)b+1 dim im ∂b+1 .
n=a+1
□
Now the rank-nullity theorem finishes the job.
... Corollary. e Euler characteristic of a finite CW complex does not depend upon the chosen CW decomposition.
... Corollary. For any distinguished triangle E′ . E . E′′ . E′ [1] of bounded complexes of finite dimensional
rational vector spaces, one has
χ(E′ ) − χ(E) + χ(E′′ ) = 0.
Now for a pointed finite CW complex (X, x), it is convenient to modify the definition of the Euler characteristic
by subtracting 1, so that
χ(X, x) = χ(X) − 1.

e result is thus the Euler characteristic of the reduced cellular complex. Since a cofiber sequence of pointed CW
complexes induces a distinguished triangle on the reduced cellular complexes, we have the following further corollary.
... Corollary. For any pointed map f : (X, x) .
(Y, y), one has
χ(X, x) − χ(Y, y) + χ(Z, z) = 0,
where Z = Cf is the mapping cone of f.
.. Example. Here are some generalizations of the examples we’ve been contemplating.
(..) For any commutative ring R, denote by Perf(R) the category of bounded complexes of finitely generated
projective A-modules. If R is a domain with fraction field K, then for any E ∈ Perf(R), one can define an
integer
∑
χ(E) =
(−1)n dimK (En ⊗R K);
n∈Z
this defines a map χ : Obj Perf(R) . Z.
(..) For any space B, denote by CWB/ /B the category of finite relative CW complexes (X, B) equipped with a
retraction X . B. We may define a map χ : CWB/ /B . Z given by
∑
χ(X, B) =
(−1)n dim Hn (X, B; Q).
n≥0
In both of these examples we have attached to any object of a suitable category C a number, in such a way as to
guarantee that “all cofibration sequences split.” is idea makes sense even if the thing we attach to an object of C is
not a number, but an element of an abelian group.
.. Definition. Suppose C a small pointed category (i.e., a category with a zero object — an object that is both initial
and terminal), and suppose C “admits a good theory of homotopy cofibers.” Suppose A an abelian group. en a
generalized Euler characteristic valued in A is a map χ : Obj C . A satisfying the following additivity constraint: for
any cofiber sequence
U. V. W
in C , one has χ(U) − χ(V) + χ(W) = 0.
In due course, we will explain carefully what we mean when we say that C must “admit a good theory of homotopy
cofibers.” For now, let us simply comment that virtually all of the “pointed” categories from topology will be of this
type. In particular, all the examples above are generalized Euler characteristics.
.. Example (Knudsen–Mumford, []). Let R be a commutative ring. en for any E ∈ Perf(R), we may speak of
the determinant of E as an alternating tensor product:
⊗
n
det(E) =
(det En )(−1) ,
n∈Z
which we may regard as an element of the Picard group Pic(R) of invertible R-modules. It can be shown that for any
distinguished triangle
E′ . E . E′′ ,
one has det(E) ∼
= det(E′ ) ⊗ det(E′′ ), whence the map det : Obj Perf(R) . Pic(R) is a generalized Euler characteristic.
For any abelian group A and any small pointed category C that “admits a good theory of homotopy cofibers,” let
us write X(C , A) for the set of Euler characteristics Obj C . A. It is clear that for any homomorphism φ : A′ . A
of abelian groups, composition with φ defines a map X(C , A′ ) . X(C , A). Hence X(C , −) is a functor Ab . Set.

.. eorem. e functor X(C , −) is corepresentable; that is, there exist an abelian group K0 (C ) and an isomorphism
X(C , A) ∼
= Hom(K0 (C ), A),
functorial in A.
Proof . We can use the representability requirement. It is easy to check that X(C , −) preserves all products and
pullbacks, and hence with respect to all limits. If we choose an uncountable regular cardinal κ that is larger than the
cardinality of Obj C , then it is easy to see that X(C , −) preserves all κ-filtered unions of abelian groups.
□
Proof . Alternately, we can construct explicitly the representing object. Form the free abelian group F(C ) on the
objects of C , and let F0 (C ) be the subgroup thereof generated by the set
{U − V + W | U, V, W ∈ Obj C , and there exists a cofiber sequence U .
V.
W}.
□
Now set K0 (C ) = F(C )/F0 (C ).
is means that there is an essentially unique uniersal generalized Euler characteristic: a generalized Euler characteristic
χ uni : Obj C . K0 (C )
such that for any generalized Euler characteristic χ : Obj C . A, there exists a unique homomorphism of abelian
groups φ : K0 (C ) . A with the property that χ = φ ◦ χ uni .
e 0 (R) such that one has a splitting
.. Proposition. For any domain R, there exists an abelian group K
e 0 (R).
K0 (Perf(R)) ∼
=Z⊕K
Proof. For any abelian group A, define
as the assignment χ .
η : X(Perf(R), A) .
χ(R); in the opposite direction, define
θ: A .
as the assignment α .
χ α , where χ α is defined by
χ α (E) =
A
X(Perf(R), A)
∑
(−1)n (dim En )α.
n∈Z
We see instantly that η ◦ θ = idA . Since η and θ are both natural in A, it follows that the functor X(Perf(R), −)
splits as id ⊕F for some functor F : Ab . Set.
□
.. Example. For any field k, one has K0 (k) ∼
= Z.
Proof. With the maps η and θ as above, we must show that θ ◦ η = id. at is, we must show that any generalized
Euler characteristic χ : Perf(k) . A must satisfy
∑
(−1)n (dim En )χ(k).
χ(E) =
n∈Z
If E is concentrated in degree zero, then it is isomorphic to kn for some n, whence by additivity χ(E) = nχ(k).
Considering the distinguished triangle E . 0 . E[1] . E[1] leads one to deduce that χ(E[m]) = (−1)m χ(E).
To complete the proof, we may induct on the length of E in the obvious manner.
□
.. Exercise. Show that if R is a Dedekind domain, then in fact the generalized Euler characteristic
χ ⊕ det : Obj Perf(R) . Z ⊕ Pic(R)
∼
exhibits an isomorphism K0 (Perf(R))√
= Z ⊕ Pic(R). Use the identification of Pic(R) with the ideal class group to
compute the groups K0 (Z) and K0 (Z[ d] for as many squarefree integers as you can.
.. Example. e Euler characteristic χ(−, B) in fact exhibits an isomorphism K0 (CWB/ /B ) ∼
= Z.

Proof. Let us verify this in the case where B = ∗. Now the rational reduced homology functor
C∗ : CW∗ .
Perf(Q)
induces a natural transformation θ : id ∼
= X(Perf(Q), −) . X(CW∗ , −). On the other hand the assignment
χ . χ(S0 ) induces a natural transformation η : X(CW∗ , −) . id. It is easy to see that η ◦ θ = id. In the other
direction, we have to show that for any generalized Euler characteristic χ : Obj CW∗ . A, one has
∑
χ(X, x) =
(−1)n (dim Hn (X, x; Q))χ(S0 ).
n≥0
For this, we note that the cofiber sequence Y . 0 . ΣY guarantees that χ(ΣY) = (−1)χ(Y), and a cell attachment
n−1
Y′ = Dn ∪S Y along a map Sn−1 . Y gives a cofiber sequence Sn−1 . Y . Y′ , whence χ(Y′ ) = χ(Y) −
χ(Sn−1 ) = χ(Y) + (−1)n χ(S0 ). e result thus follows by induction.
□
Next time, we’ll say more about what sort of categories are suitable for K0 , and we’ll start to see that the group K0
is part of a much richer structure — the K-theory space — that we will want to characterize in a manner similar to
our characterization of K0 .
R
. Finn Faye Knudsen and David Mumford, e projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand.
 (), no. , –. MR  ( )
