Connective complex K-theory spectrum

For multiplicative data use simplicial symmetric spectra, with the following conventions. A pointed simplicial set is a functor from the opposite of the category of finite nonempty linearly ordered sets and order-preserving maps to pointed sets. Write $\mathrm{\Delta }[d{]}_m={\mathrm{Hom}}_{\mathrm{o}\mathrm{r}\mathrm{d}}([m],[d])$, with $[d]=\{0,\dots ,d\}$, and let $\mathrm{\partial }\mathrm{\Delta }[d]$ be its union of proper faces. Put $S^1=\mathrm{\Delta }[1]\mathrm{/}\mathrm{\partial }\mathrm{\Delta }[1]$ and $S^n=(S^1{)}^{\wedge n}$, with $S^0$ the two-point unit; $\wedge$ is the pointed product modulo the wedge. A symmetric sequence $E$ consists of pointed simplicial sets $E_n$, $n\ge 0$, with basepoint-preserving actions of the permutation groups ${\mathrm{\Sigma }}_n$. Its tensor product is $(E\otimes F{)}_n={\bigvee }_{p+q=n}({\mathrm{\Sigma }}_n{)}_{+}{\wedge }_{{\mathrm{\Sigma }}_p\times {\mathrm{\Sigma }}_q}(E_p\wedge F_q)\mathrm{.}$ Here the subscript means the orbit quotient for the block-permutation action, and $+$ adjoins a disjoint basepoint. The symmetry exchanges the two factors and their coordinate blocks. The sphere sequence $S=(S^n{)}_{n\ge 0}$ has multiplication by concatenation of sphere coordinates. A symmetric spectrum is a unital associative left $S$-module: a symmetric sequence with equivariant maps $S^p\wedge E_q\to E_{p+q}$ satisfying the unit and iterated-action identities. Its structure maps are the cases $p=1$; morphisms preserve the actions. Define $E{\wedge }_{S}F=\mathrm{coeq}(E\otimes S\otimes F\rightrightarrows E\otimes F),$ using the two $S$-actions, with the right action on $E$ obtained by the symmetry. A ring spectrum here is such an $R$ with morphisms $\eta :S\to R$ and $\mu :R{\wedge }_{S}R\to R$ satisfying $\mu (\mu {\wedge }_{S}1)=\mu (1{\wedge }_S\mu )$ and $\mu (\eta {\wedge }_{S}1)=1=\mu (1{\wedge }_S\eta )$, with the canonical associativity and unit identifications. A map of ring spectra preserves both $\eta$ and $\mu$.

Specify also the category in which stable brackets are taken. A weak equivalence of simplicial sets means that geometric realization induces a bijection of path components and isomorphisms of all homotopy groups at every basepoint. A level equivalence of symmetric spectra is a weak equivalence at every level. A symmetric Omega-spectrum has Kan levels (each map from the union of all but one face of $\mathrm{\Delta }[d]$ extends over $\mathrm{\Delta }[d]$, for $d\ge 1$) and weak-equivalence adjoints $E_n\to {\mathrm{Map}}_{\ast }(S^1,E_{n+1})$. Call $T$ injective if every map $A\to T$ extends over each levelwise injective level equivalence $A\to B$. Put $\mathrm{Map}(E,T{)}_d=\mathrm{Hom}(E\wedge \mathrm{\Delta }[d{]}_{+},T)$, where the tensor with a pointed simplicial set is levelwise. A stable equivalence $f:E\to F$ means that ${\pi }_0\mathrm{Map}(F,T)\to {\pi }_0\mathrm{Map}(E,T)$ is bijective for every injective symmetric Omega-spectrum $T$. Define $\mathrm{S}\mathrm{H}={\mathrm{S}\mathrm{p}}^{\mathrm{\Sigma }}[W^{-1}]$, where $W$ is this class: formally adjoin inverses to its maps and impose the category and inverse relations. Brackets $[E,F]$ mean ${\mathrm{Hom}}_{\mathrm{S}\mathrm{H}}(E,F)$. In particular, the equivalences used in these brackets are specified, rather than inferred from naive stable groups of arbitrary symmetric spectra.

For a based CW space $X$, its suspension spectrum is the symmetric spectrum $({\mathrm{\Sigma }}^{\mathrm{\infty }}X{)}_n=S^n\wedge {\mathrm{Sing}}_{\ast }X$, where pointed singular simplices are based at the constant basepoint simplex. Write ${\mathrm{\Sigma }}^{q}E$ for the stable suspension shift: for $q\ge 0$ it is $S^q\wedge E$ levelwise, and for $q=-a<0$ use levelwise ${\mathrm{Map}}_{\ast }(S^a,E^{\mathrm{\Omega }})$, with the induced structure maps, where a functorial stable-equivalence replacement $E\to E^{\mathrm{\Omega }}$ with Omega-spectrum target is fixed as part of the model data. For these data compute ${\pi }_j(E)$ from the underlying realized sequential Omega-prespectrum $\mathrm{\mid }{E}^{\mathrm{\Omega }}\mathrm{\mid }$ by ${\mathrm{colim}}_n{\pi }_{n+j}(\mathrm{\mid }{E}_n^{\mathrm{\Omega }}\mathrm{\mid })$, starting with $n+j\ge 2$. Thus the earlier sequential-prespectrum definitions specify the stable-group calculation; the symmetric structure supplies the multiplication and stable mapping convention.

Fix the periodic complex $K$-theory ring spectrum $KU$, together with its Bott underlying Omega-prespectrum model $K{U}_{2r}=\mathbb{Z}\times BU$ and $K{U}_{2r+1}=U$ for $r\ge 0$. Here $U={\mathrm{colim}}_{s}U(s)$ under $a\mapsto \mathrm{diag}(a,1)$ and $BU={\mathrm{colim}}_s{\mathrm{Gr}}_s({\mathbb{C}}^{2s})$ under addition of a fixed complex line; take well-pointed CW models, based at the identity and the zero virtual bundle. The adjoint structure maps are the bundle-clutching equivalence $U\to \mathrm{\Omega }(\mathbb{Z}\times BU)$ and the Bott equivalence $\mathbb{Z}\times BU\to \mathrm{\Omega }U$, normalized by the Bott generator. These displayed spaces specify an underlying Omega-model, rather than extra permutation actions on those very spaces. The chosen ring multiplication induces stable-bundle tensor product. The representing convention is $K{U}^q(Y,B)=[{\mathrm{\Sigma }}^{\mathrm{\infty }}(Y\mathrm{/}B),{\mathrm{\Sigma }}^{q}KU]$ for a finite CW pair, identified with its complex $K^q(Y,B)$, with $Y\mathrm{/}\mathrm{\varnothing }=Y_{+}$.

A connective cover datum for $KU$ is a map $i:k\to KU$ of ring spectra, with the above choices of underlying Omega-models, such that ${\pi }_j(k)=0$ for $j<0$ and ${\pi }_j(i)$ is an isomorphism for $j\ge 0$. Fix the standard connective cover used in the source model, together with its complexified Chern character $c:k\to H\mathbb{C}[u]$, where $\mathbb{C}[u]$ has zero differential and $u$ has homological degree $2$, and $H$ denotes its multiplicative Eilenberg–Mac Lane spectrum. For any CW space $Y$ and $q\in \mathbb{Z}$, put $k^q(Y)=[{\mathrm{\Sigma }}^{\mathrm{\infty }}{Y}_{+},{\mathrm{\Sigma }}^{q}k]$. These are topological spectrum data, prior to any differential refinement.

Complex K-theory is a two-periodic generalized cohomology theory

Stable homotopy groups of a sequential prespectrum

Sequential prespectra, spectra, and adjoint structure maps

Complex Bott periodicity