Corollary 9.4.3. The underlying anima of \(\BUP \) is \(\Z \times \bB U\): \[ \BUP \simeq (\Vect (\pt )^{\simeq })^{\grp } \simeq \Z \times \bB U. \]
Proof. By Proposition 9.4.1, the commutative monoid \(M=\Vect (\pt )^{\simeq }\) has underlying anima \(\bigsqcup _{k\geq 0}\bB U(k)\), with addition induced by block sum. Let \(x\in M\) be the point corresponding to the one-dimensional vector space \(\C \). The telescope \(T(M,x)\) is obtained by repeatedly adding this line. Its component of virtual rank \(m\in \Z \) is the colimit \[ \colim _{n\geq \max (0,-m)} \bB U(m+n), \] which is canonically \(\bB U\). Thus \(T(M,x)\simeq \Z \times \bB U\).
The two hypotheses of Theorem 5.5.2 are immediate. The class of \(x\) is the generator \(1\in \pi _0(M)\cong \N \). The fundamental group of \(\bB U\) is \(\pi _0(U)\), which is trivial because the stable unitary group is connected. Hence \(T(M,x)\to M^{\grp }\) is an isomorphism. □
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