Section 16.4 Group Completion

From this point onward we work with complex vector bundles. By local triviality the rank of a bundle is locally constant, so it is constant on each connected component of \(X\text{.}\) When \(X\) is not connected we allow the rank to differ between components, and write \(\mathrm{Vect}(X)\) for the set of isomorphism classes of such bundles. For connected \(X\) this is simply \(\mathrm{Vect}(X) = \bigsqcup_{k \geq 0} \mathrm{Vect}_k(X)\text{.}\)

Direct sum gives a commutative and associative binary operation

\begin{equation*} \oplus \colon \mathrm{Vect}(X) \times \mathrm{Vect}(X) \to \mathrm{Vect}(X), \qquad ([E], [E']) \mapsto [E \oplus E'], \end{equation*}

making \(\mathrm{Vect}(X)\) a commutative monoid with unit \([\varepsilon^0]\text{,}\) the rank-\(0\) trivial bundle. A cohomology theory takes values in abelian groups, because exactness is a statement about kernels and images. There are no inverses in \(\mathrm{Vect}(X)\text{,}\) since rank only increases under \(\oplus\text{.}\) So we add inverses formally, in the most efficient way possible.

Proposition 16.10. Group completion.

The forgetful functor \(U \colon \mathbf{Ab} \to \mathbf{CMon}\) from abelian groups to commutative monoids has a left adjoint

\begin{equation*} (-)^{\mathrm{gp}} \colon \mathbf{CMon} \to \mathbf{Ab}, \end{equation*}

called group completion (or the Grothendieck group of the monoid).

Concretely, the adjunction says that there is a monoid map \(\iota \colon M \to M^{\mathrm{gp}}\) such that every monoid map \(\varphi \colon M \to A\) to an abelian group factors uniquely as \(\varphi = \bar\varphi \circ \iota\) with \(\bar\varphi\) a group homomorphism.

Proof.

Think of a pair \((m_1, m_2) \in M \times M\) as the formal difference “\(m_1 - m_2\)”. Define \(M^{\mathrm{gp}} := (M \times M)/\!\sim\text{,}\) where \((m_1, m_2) \sim (n_1, n_2)\) if there exists \(k \in M\) with

\begin{equation*} m_1 + n_2 + k \;=\; m_2 + n_1 + k. \end{equation*}

Without the extra summand \(k\) this would be the usual rule for equality of differences. The summand is needed because \(M\) need not have cancellation, and without it the relation can fail to be transitive. Componentwise addition makes \(M^{\mathrm{gp}}\) a commutative monoid, and it is a group because \((m_1, m_2) + (m_2, m_1) = (m_1 + m_2, m_1 + m_2) \sim (0,0)\text{.}\) Set \(\iota(m) = [m, 0]\text{.}\)

Given \(\varphi \colon M \to A\text{,}\) define \(\bar\varphi[m_1, m_2] := \varphi(m_1) - \varphi(m_2)\text{.}\) This is well defined. Indeed, applying \(\varphi\) to \(m_1 + n_2 + k = m_2 + n_1 + k\) and cancelling \(\varphi(k)\) in the group \(A\) gives \(\varphi(m_1) - \varphi(m_2) = \varphi(n_1) - \varphi(n_2)\text{.}\) It is a homomorphism with \(\bar\varphi \circ \iota = \varphi\text{.}\) It is unique, because \([m_1, m_2] = \iota(m_1) - \iota(m_2)\) shows that \(\iota(M)\) generates \(M^{\mathrm{gp}}\text{.}\)

Example 16.11.

\(\bigl(\mathbb{N}, +\bigr)^{\mathrm{gp}} = \mathbb{Z}\text{,}\) with \([m_1, m_2] \mapsto m_1 - m_2\text{.}\)

Example 16.12.

\(\bigl(\mathbb{N}_{>0}, \times\bigr)^{\mathrm{gp}} = \mathbb{Q}_{>0}\text{,}\) with \([m_1, m_2] \mapsto m_1/m_2\text{.}\) By unique factorisation this is the free abelian group on the primes.

Example 16.13.

The map \(\iota \colon M \to M^{\mathrm{gp}}\) need not be injective. It identifies \(m\) and \(n\) exactly when \(m + k = n + k\) for some \(k\text{.}\) In the extreme case of \(M = \mathbb{N} \cup \{\infty\}\) under addition, \(m + \infty = \infty = n + \infty\) for all \(m, n\text{,}\) so \(M^{\mathrm{gp}} = 0\text{.}\) For vector bundles this means that \(K(X)\) only sees bundles up to adding a common summand. We make this precise in Remark 16.17.

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