Complex K-theory spectra

Proof. By Definition 19.7.8, Proposition 19.7.9, the rank \(n\) component of \(\Vect (\pt )^{\simeq }\) is the geometric realization of the simplicial groupoid whose value at \([q]\) is the full subgroupoid \[ \left (\Vect ^{\disc }(\abs {\Delta ^q})^{\simeq }\right )_{\rk =n} \] spanned by the rank \(n\) bundles. Every rank \(n\) vector bundle over \(\abs {\Delta ^q}\) is trivial. Consequently, the full subgroupoid on the standard trivial bundle \(\abs {\Delta ^q}\times \C ^n\) is an equivalent one-object groupoid. Its automorphism group is \[ \Hom _{\Top }(\abs {\Delta ^q},\GL _n(\C )), \] and these groups assemble into the singular simplicial group of \(\GL _n(\C )\). Geometric realization preserves finite products by Lemma 21.6.12, so it commutes with the formation of classifying animae of simplicial groups. Since the realization of the singular simplicial group is the underlying group anima of \(\GL _n(\C )\), the realization of these one-object groupoids is \(\bB \GL _n(\C )\). Taking the disjoint union over all ranks therefore gives \[ \Vect (\pt )^{\simeq } \simeq \bigsqcup _{n \geq 0} \bB \GL _n(\C ). \] The inclusions \(U(n) \hookrightarrow \GL _n(\C )\) are homotopy equivalences by polar decomposition and commute with block sums and tensor products. They therefore identify \(\bB \GL _n(\C )\) with \(\bB U(n)\) compatibly with both operations.

Indeed, direct sum sends \((\C ^k,\C ^l)\) to \(\C ^{k+l}\) and induces the block-sum inclusion on automorphism groups. Similarly, tensor product sends \((\C ^k,\C ^l)\) to \(\C ^k\otimes \C ^l\cong \C ^{kl}\) and induces the tensor-product representation on automorphism groups. Passing to classifying animae gives the stated formulas. □

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. □

Proof. The space \(\Z \times BU\) is a cell complex by the Schubert cell description above. Choose a homotopy equivalence \(X'\to X\) with \(X'\) a cell complex. Then \[ [X,\Z \times BU]\cong [X',\Z \times BU] \cong \pi _0\Hom _{\An }(\Pi _\infty X',\Pi _\infty (\Z \times BU)) \cong \pi _0\Hom _{\An }(\Pi _\infty X,\Pi _\infty (\Z \times BU)), \] where the middle isomorphism is Corollary 2.3.7. Combining this with Proposition 9.3.8, Proposition 9.2.21, Corollary 9.4.3 and the adjunction \(\S [-]\dashv \Omega ^{\infty }\) gives \begin {align*} K^0(X) &\cong [X,\Z \times BU] \\ &\cong \pi _0\Hom _{\An }(\Pi _{\infty }(X),\Pi _{\infty }(\Z \times BU)) \\ &\cong \pi _0\Hom _{\An }(\Pi _{\infty }(X),\Z \times \bB U) \\ &\cong \pi _0\Hom _{\An }(\Pi _{\infty }(X),\BUP ) \\ &\cong \pi _0\Hom _{\Sp }(\S [\Pi _{\infty }(X)],\ku ) \\ &= \ku ^0(\Pi _{\infty }(X)). \end {align*}

It remains to check multiplication. Set \(M:=\Vect (\pt )^{\simeq }\). The unit map \(M\to M^{\grp }=\BUP \) is a morphism of commutative semiring objects, since group completion is symmetric monoidal by Proposition 16.6.4. For a vector bundle \(E\) over \(X\), its classifying map to \(M\) therefore determines a class in \(\ku ^0(\Pi _\infty X)\), and Proposition 9.4.1 shows that direct sum and tensor product of bundles map to the sum and product of the corresponding classes. This semiring map \[ \pi _0\Vect (X)\longrightarrow \ku ^0(\Pi _\infty X) \] extends uniquely to a ring map from \(K^0(X)\). Under the description \(M^{\grp }\simeq \Z \times \bB U\) of Corollary 9.4.3, this extension is the isomorphism displayed above: an honest vector bundle is sent to its stable classifying map. Hence that isomorphism is multiplicative. □

Proof. Apply Theorem 9.3.10 to \(X=S^n\). Under the comparison above, \[ \pi _n\ku \cong \widetilde \ku ^0(S^n) \cong \widetilde K^0(S^n), \] the map \(b_*\) is external multiplication by the Bott class. The claim therefore follows from classical Bott periodicity. □

Proof. By Proposition 19.7.11, the spectrum \(\ku \) is a commutative ring spectrum, so \(\beta \in \pi _2\ku \) is a homogeneous element of the graded-commutative ring \(\pi _*\ku \). Let \(S := \{1,\beta ,\beta ^2,\dots \}\) be the multiplicative subset it generates.

By Corollary 8.4.5, the localization \(\ku [\beta ^{-1}] := \ku [S^{-1}]\) exists in \(\CAlg (\Sp )\), and the localization map \(\eta \colon \ku \to \ku [\beta ^{-1}]\) is a morphism of commutative ring spectra. By Lemma 8.4.6, its underlying spectrum is the telescope \[ \colim \big ( \ku \xrightarrow {\ b\ } \ku [-2] \xrightarrow {\ b[-2]\ } \ku [-4] \to \cdots \big ), \] where \(b\) is multiplication by \(\beta \), i.e. the Bott map of Construction 9.4.5. This colimit is precisely the spectrum \(\KU \) of Definition 9.4.7, and under this identification \(\eta \) is the canonical map \(\iota \colon \ku \to \KU \). □

Proof. By Lemma 4.4.28, the functor \(\pi _k\) preserves the filtered colimit defining \(\KU \). Moreover \(\pi _k(\ku [-2j])=\pi _{k+2j}\ku \), and the transition maps are induced by the Bott map. □

Proof. For \(k \geq 0\), Lemma 9.4.9 writes \(\pi _k\KU \) as a filtered colimit whose transition maps \[ b_*\colon \pi _{k+2j}\ku \to \pi _{k+2j+2}\ku \] are all isomorphisms by Corollary 9.4.6. Hence \(\pi _k(\iota )\) is an isomorphism for \(k\geq 0\). Since \(\ku \) is connective, the map \(\iota \) factors through the connective cover as \[ \ku \xrightarrow {\bar \iota } \tau _{\geq 0}\KU \to \KU \] by Corollary 5.4.7. The map \(\bar \iota \) is an isomorphism on all homotopy groups, because both source and target are connective and we have already checked the non-negative homotopy groups. It is therefore an equivalence by Corollary 4.4.27. □

Proof. For every connective spectrum \(Y\), the equivalence \(\ku \simeq \tau _{\geq 0}\KU \) induces an equivalence \[ \Hom _{\Sp }(Y,\ku ) \iso \Hom _{\Sp }(Y,\KU ). \] Apply this to \(Y=\Sigma ^{-k}\S [X]\), which is connective for \(k\leq 0\). □

Proof. The spectrum \(\KU [-2]\) is the colimit of the diagram defining \(\KU \) with its first term omitted: \[ \ku [-2] \xrightarrow {b[-2]} \ku [-4] \to \cdots . \] This subdiagram is cofinal in the original sequential diagram, so \(\KU [-2]\simeq \KU \). Applying \(\Sigma ^2\) gives the desired equivalence. □

Proof. The description \(\BUP \simeq \Z \times \bB U\) of Corollary 9.4.3 gives \(\pi _0\ku \cong \Z \) and \(\pi _1\ku =0\). Multiplication by \(\beta \) induces isomorphisms \(\pi _n\ku \to \pi _{n+2}\ku \) for \(n\geq 0\) by Corollary 9.4.6. Since \(\ku \) is connective, this proves \(\pi _*\ku \cong \Z [\beta ]\). The calculation for \(\KU =\ku [\beta ^{-1}]\) now follows from Lemma 7.2.4. □

Proof. By the smashing description of rationalization and Theorem 7.2.14, we have \(\KU _{\Q }\simeq \KU \otimes H\Q \), so \(\KU _{\Q }\) is canonically a commutative \(H\Q \)-algebra. Under the symmetric monoidal equivalence \(\D (\Q )\simeq \Mod _{H\Q }\) of Corollary 8.3.4, the polynomial algebra \(\Q [\beta ]\) is the free commutative \(\Q \)-algebra on a generator in degree \(2\). The rationalized Bott class therefore determines a morphism of commutative \(H\Q \)-algebras \[ H\Q [\beta ]\longrightarrow \KU _{\Q }. \] The canonical map \(H\Q [\beta ]\to H\Q [\beta ^{\pm 1}]\) exhibits its target as \((H\Q [\beta ])[\beta ^{-1}]\): indeed, the induced map from this localization is an isomorphism on homotopy groups by Lemma 8.4.6. Since the Bott class is invertible in \(\pi _*(\KU _{\Q })\), the preceding morphism therefore extends uniquely to \(H\Q [\beta ^{\pm 1}]\). It induces the isomorphism \[ \Q [\beta ^{\pm 1}]\xrightarrow {\ \cong \ }\pi _*(\KU _{\Q }) \] from Corollary 9.4.14 and Lemma 7.2.4, and hence is an isomorphism of commutative ring spectra. □

Proof. Since \(X\) is finite, the functor \(\hom (\S [X],-)\colon \Sp \to \Sp \) commutes with rationalization and with direct sums. Applying it to Proposition 9.4.15 gives \[ \hom (\S [X],\KU )_{\Q } \simeq \hom (\S [X],\KU _{\Q }) \simeq \bigoplus _{n\in \Z }\hom (\S [X],H\Q [2n]). \] The resulting map is induced by the Chern character of Definition 9.4.16. Since \(X\) is finite, only finitely many of the groups \(H^{k+2n}(X;\Q )\) are nonzero, so the product in its definition agrees with the direct sum above. Passing to \(\pi _{-k}\) gives the claim. □