Graded Chern character by suspension and Bott periodicity

Assume AC. All based finite CW complexes have a vertex as basepoint. Write ${\tilde{H}}^m(X;\mathbb{Q})=H^m(X,\ast ;\mathbb{Q})$; the natural map to $H^m(X;\mathbb{Q})$ identifies this with the kernel of restriction to the basepoint. Indeed restriction is split by the map to a point, so the ordinary pair sequence gives exactly that kernel, also in degree zero. Negative ordinary cohomology groups are zero.

By Reduced complex K-theory, ${\tilde{K}}^0(X)$ is likewise the kernel of restriction to the chosen basepoint. Define ${\stackrel{}{\mathrm{ch}}}^0:{\tilde{K}}^0(X)⟶{⨁}_{k\ge 0}{\tilde{H}}^{2k}(X;\mathbb{Q})$ as the restriction of Chern character is a natural ring homomorphism on K-zero. Naturality makes its image lie in the indicated kernel. Explicitly, for $a=[E]-[F]$ with $\dim E_{\ast }=\dim F_{\ast }$, its value is $\mathrm{ch}(E)-\mathrm{ch}(F)$, interpreted in that kernel. Only the rank at the chosen basepoint is required to vanish: the degree-zero component on another component of $X$ is the virtual rank there. For example, the generator supported at the nonbasepoint of $S^0$ maps to its degree-zero indicator function. No subtraction of a componentwise rank function is made.

Use the cohomological suspension isomorphism in the direction $s:{\tilde{H}}^m(X;\mathbb{Q})\stackrel{\cong }{\to }{\tilde{H}}^{m+1}(\mathrm{\Sigma }X;\mathbb{Q})\mathrm{.}$ It is the ordinary cone boundary, with the sphere coordinate first, and is natural for based maps. This construction only uses singular cohomology, not a correspondence theorem for arbitrary generalized theories: the reduced cone is contractible, its base inclusion is a CW cofibration, and excision identifies the relative cone group $H^{m+1}(CX,X;\mathbb{Q})$ with ${\tilde{H}}^{m+1}(CX\mathrm{/}X;\mathbb{Q})$. The reduced cone pair sequence makes its boundary an isomorphism, because the reduced groups of the cone vanish. These are the pair exactness, homotopy and excision clauses of Singular cohomology satisfies the Eilenberg Steenrod cohomology axioms, with cone and quotient as in Reduced cone suspension and cofiber sequence. The same construction at the one-point complex has zero source and target.

The odd character is ${\stackrel{}{\mathrm{ch}}}^{-1}:{\tilde{K}}^{-1}(X)={\tilde{K}}^0(\mathrm{\Sigma }X)\stackrel{{\stackrel{}{\mathrm{ch}}}^0}{\to }{⨁}_k{\tilde{H}}^{2k}(\mathrm{\Sigma }X;\mathbb{Q})\stackrel{s^{-1}}{\to }{⨁}_k{\tilde{H}}^{2k-1}(X;\mathbb{Q}),$ where the K-group identification is Negative-degree complex K-groups. Thus this is degree $-1$; degree $1$ is its Bott translate. In all cohomology sums the integer index ranges over exactly the degrees displayed, with negative ordinary groups zero.

Here is the periodicity normalization and the reason an even shift is harmless. The fixed Bott class is $\beta =[\gamma ]-1$ on $S^2={\mathbb{C}\mathbb{P}}^1$, with the tautological Hopf line of Hopf-line calculation of K⁰(S²). Put $u=c_1(\gamma )\in {\tilde{H}}^2(S^2;\mathbb{Z})$. It is the tautological Euler generator by The complex tautological Euler class restricts to the projective-fiber generator and the line convention of Chern character of a complex vector bundle. Since $S^2$ has no cohomology above degree two, ${\stackrel{}{\mathrm{ch}}}^0(\beta )=u_{\mathbb{Q}}\mathrm{.}$ In particular ${\tilde{K}}^0(S^2)=K^{-2}(\ast )$, not ${\tilde{K}}^{-2}(\ast )$ (the latter is zero).

On a smash product, the reduced external-product formula is ${\stackrel{}{\mathrm{ch}}}^0(a⊠\beta )={\stackrel{}{\mathrm{ch}}}^0(a)⊠u_{\mathbb{Q}}\mathrm{.}$ To justify descent from the degree-zero product formula, pull back along $X\times S^2\to X\wedge S^2$ and use External product in complex K-theory. Quotient pullback in reduced ordinary cohomology is injective: restriction from the product to its wedge of axes is surjective, with a splitting given by the two projections, in every degree; its pair sequence therefore identifies relative cohomology with the kernel of restriction. Excision identifies that relative group with the reduced cohomology of the smash quotient. Thus equality after pullback proves the displayed formula. Relative Künneth Relative cohomological Kunneth under finite free homology hypotheses shows that external product with $u_{\mathbb{Q}}$ is an isomorphism shifting ordinary cohomological degree by two: the reduced homology of the sphere is one copy of $\mathbb{Q}$ in degree two. Use precisely this generator and this two-suspension identification when matching Bott periodicity; replacing $u$ by the oppositely oriented generator requires the corresponding sign change in that identification.

For any integer $j$, choose an even shift $2r$ making $j-2r$ equal to $0$ or $-1$ and use the fixed Bott isomorphism $B^r:K^j(X)\to K^{j-2r}(X)$, allowing negative powers of $B$. Apply the corresponding character above and reindex the ordinary groups as $H^{j+2k}$. The preceding product identity, also applied to $\mathrm{\Sigma }X$, shows that inserting another Bott step and its cohomological two-suspension identification gives the same map. Iteration proves independence of any larger nonpositive suspension representative; the inverses obey the same identity. The Bott isomorphisms and their inverses are those of Complex Bott periodicity. This defines natural additive maps ${\mathrm{ch}}^j:K^j(X)⟶{⨁}_{k\in \mathbb{Z}}{H}^{j+2k}(X;\mathbb{Q})\mathrm{.}$ For unbased $X$ use $X_{+}$ in the reduced construction, as prescribed in Negative-degree complex K-groups; thus ${\tilde{H}}^m(X_{+};\mathbb{Q})=H^m(X;\mathbb{Q})$. This includes the empty unbased space, for which both sides are zero. Every sum is finite because $X$ is finite dimensional.

The maps use the external products and suspension conventions of the multiplicative theory Complex K-theory is a two-periodic generalized cohomology theory. Compatibility with external products is checked by suspending each factor into degree zero, using the degree-zero external formula, and desuspending. The permutation bringing the sphere coordinates together is the same on both sides; its Koszul sign is the one in the graded external product Cohomological Kunneth cross product is a ring isomorphism. Finally the displayed Bott-product identity permits the same transport in positive degrees. Thus product compatibility uses the prescribed coherent suspension products as well as the Bott normalization; periodicity of the source groups alone would not establish it.

Hatcher, Vector Bundles & K-Theory, §4.1, printed pp.110–111, defines the reduced character by the kernels of basepoint restriction, proves the Bott/external-product square in Proposition 4.3, and defines the odd character by the suspension square immediately before Proposition 4.5. The generator here is explicitly $c_1(\gamma )$, preserving this page's existing tautological-line convention.