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Smooth $$K$$-theory. (English) Zbl 1202.19007
Dai, Xianzhe (ed.) et al., From probability to geometry II. Volume in honor of the 60th birthday of Jean-Michel Bismut. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-289-1/pbk). Astérisque 328, 45-135 (2009).
Given a smooth $$n$$-manifold $$M$$ and a proper subring $$\Lambda$$ of $$\mathbb R$$, the smooth cohomology group $$\widehat H^{k+1}(M;\Lambda)$$ is the group of homomorphisms $$\chi$$ from the group of smooth singular $$k$$-cycles $$Z_k(M;\mathbb R)$$ into $$\mathbb R/\Lambda$$ with the property that there exists a real $$(k+1)$$-form $$\omega\in\Omega^{k+1}(M)$$ such that $$\chi(\partial c)=\int_c\omega+\Lambda$$ for every smooth singular $$(k+1)$$-chain $$c\in C_{k+1}(M;\mathbb R)$$. The relations with usual cohomology are as follows.
A nonvanishing real form never takes values in a proper subring of $$\mathbb R$$; consequently $$\omega$$ must be closed, and $$R:\widehat H^k(M;\Lambda)\to\Omega^k_{\text{cl}}(M)=\ker d\i\Omega^k(M)$$, $$\chi\mapsto\omega$$, is well-defined. It turns out that $$\operatorname{im} R$$ equals the kernel of the composition $$\Omega^k_{\text{cl}}(M)\to H^k(M;\mathbb R)\to H^k(M;\mathbb R/\mathbb Z)$$, and $$\ker R$$ is isomorphic to $$H^{k-1}(M;\mathbb R/\mathbb Z)$$. On the other hand, since $$\mathbb R$$ is divisible, every $$\chi$$ as above lifts to a homomorphism $$Z_k(M;\mathbb R)\to\mathbb R$$, which in turn is the restriction of a real singular cochain $$\phi\in C^k(M;\mathbb R)$$. Since $$\omega$$ is closed and well-defined, the cochain $$\psi\in C^{k+1}(M;\Lambda)$$, $$\psi(c)=\delta\phi(c)-\int_c\omega$$, is a cocycle, and $$I:\widehat H^{k+1}(M;\Lambda)\to H^{k+1}(M;\Lambda)$$, $$\chi\mapsto[\psi]$$, is well-defined.
It turns out that $$I$$ is an epimorphism, and its kernel is isomorphic to the cokernel of the composition $$H^k(M;\Lambda)@>c>>H^k(M;\mathbb R)\i\Omega^k(M)/\operatorname{im} d$$ [J. Cheeger and J. Simons, Geometry and topology, Proc. Spec. Year, College Park/Md. 1983/84, Lect. Notes Math. 1167, 50-80 (1985; Zbl 0621.57010)]. A geometric description of $$\widehat H^k(M;\mathbb Z)$$ in the spirit of S. Buoncristiano, C. P. Rourke and B. J. Sanderson [A geometric approach to homology theory. London Mathematical Society Lecture Note Series. 18. Cambridge etc.: Cambridge University Press. (1976; Zbl 0315.55002)] has been given by the authors and M. Kreck [Ann. Math. Blaise Pascal 17, No. 1, 1–16 (2010; Zbl 1200.55007)].
Now let $$h$$ be a generalized (in the usual sense) cohomology theory, and let us fix a natural transformation of theories $$c:h^*(X)\to H(X;V)^*$$, where $$V$$ is a $$\mathbb Z$$-graded vector space and $$H(X;V)^k=\sum_j H^i(X;V^{k-j})$$. The principal example is as follows: $$V=h^*(pt)\otimes\mathbb R$$ and $$c$$ is the real-valued Chern–Dold character $$h^*(X)@>\text{ch}_h>>H(X;h^*(pt)\otimes\mathbb Q)^*@>i^*>> H(X;h^*(pt)\otimes\mathbb R)^*$$, that is the unique natural transformation of theories that for $$X=pt$$ is given by $$a\mapsto a\otimes 1$$. (Apart from A. Dold’s original paper [Colloq. algebr. Topology, Aarhus 1962, 2–9 (1962; Zbl 0145.20104)], a brief construction of $$\text{ch}_h$$ appears in the Hopkins–Singer paper cited below, and a more detailed treatment is found in V. M. Buchstaber’s survey [J. Sov. Math. 11, 815–921 (1979; Zbl 0428.55002)].) The authors define a “smooth extension” of $$h$$, which is a functor $$\widehat h$$ from the category of smooth manifolds and smooth maps to $$\mathbb Z$$-graded abelian groups, along with additional data:
(i)
an epimorphic natural transformation $$I\:\widehat h^*(M)\to h^*(M)$$;
(ii)
a natural equivalence $$a$$ of degree $$-1$$ between $$\ker I$$ and the cokernel of the composition $$h^*(M)@>c>>H(M;V)^*\i[\Omega(M)/\operatorname{im} d]^*\otimes_\mathbb R V$$; and
(iii)
a natural transformation $$R:\widehat h^*(M)\to\Omega_{\text{cl}}(M)^*\otimes_\mathbb R V$$ such that $$Ra^{-1}=d$$ and $$cI=qR$$, where $$q$$ is the surjection $$\Omega_{\text{cl}}(M)^*\otimes_\mathbb R V\to H(M;V)^*$$.
M. Hopkins and I. M. Singer [J. Differ. Geom. 70, No. 3, 329–452 (2005; Zbl 1116.58018)] proved that a smooth extension $$\widehat h$$ of $$(h,c)$$ always exists.
As a motivation for considering smooth extensions of generalized cohomology theories the present authors mention “the problem of setting up Lagrangians for quantum field theories with differential form field strength”.
In the paper under review, the authors use geometric structures on vector bundles and submersions, and analytic methods, to construct a smooth extension $$\widehat K$$ of complex $$K$$-theory with respect to the real-valued Chern character $$i^*\text{ch}_K$$, along with a natural pushforward $$\widehat p_!:\widehat K(E)\to\widehat K(B)$$ for a smoothly $$K$$-oriented proper submersion $$p:E\to B$$, satisfying a number of properties. They also construct a lifted Chern character $$\widehat{\text{ch}}_K:\widehat K^*(M)\to\widehat H(M;\mathbb Q)^*$$ and show that it induces a rational isomorphism.
Moreover, they construct a modified pushforward $$\widehat p^A_!:\widehat H(E;\mathbb Q)^*\to\widehat H(B;\mathbb Q)^*$$ and prove a smooth extension of the cohomological Atiyah-Singer index theorem: $$\widehat p_!^A\widehat{\text{ch}}_K= \widehat{\text{ch}}_K\widehat p_!$$.
The question of uniqueness of smooth extensions of generalized cohomology theories has been studied previously by the authors [U. Bunke and T. Schick, J. Topol. 3, No. 1, 110–156 (2010; Zbl 1252.55002); correction, arXiv:1007.2788]. In particular, they proved that complex $$K$$-theory admits infinitely many inequivalent smooth extensions with respect to $$i^*\text{ch}_K$$, but at most one such extension $$\widehat K$$ admitting a natural pushforward $$\widehat p_!:\widehat K(M\times S^1)\to\widehat K(M)$$ for the projection $$p\:M\times S^1\to M$$, where $$M$$ is an arbitrary smooth manifold, satisfying $$\widehat p_!p^*=0$$ and compatible with $$a$$, $$I$$ and $$R$$. The extension $$\widehat K$$ and the pushforward $$\widehat p_!$$ constructed in the paper under review do satisfy these properties.
The “flat” part of a smooth extension $$\widehat h$$ of a pair $$(h,c)$$, that is the kernel of the “curvature” transformation $$R$$, is a homotopy invariant functor; the flat part of the Hopkins-Singer $$\widehat h$$ is more specifically the restriction to smooth manifolds of a generalized (in the usual sense) cohomology theory $$h_{\mathbb R/\mathbb Z}$$, “the homotopy theorist’s version of $$h$$ with $$\mathbb R/\mathbb Z$$ coefficients” [the authors, J. Topology, op. cit.]. (See Buoncristiano-Rourke-Sanderson, op. cit., for a geometric topologist’s definition of $$h_{\mathbb R/\mathbb Z}$$.)
In the present paper, the authors conclude from their previous results that the even-graded part and the flat part of their $$\widehat K$$ coincide with those of the Hopkins–Singer smooth extension of complex $$K$$-theory. On the flat part $$K_{\mathbb R/\mathbb Z}$$ of $$\widehat K$$, which is further identified with the “multiplicative $$K$$-theory” $$\operatorname{Hom}(K_*(M),\mathbb R/\mathbb Z)$$, the authors’ pushforward $$\widehat p_!$$ and lifted Chern character $$\widehat{\text{ch}}_K$$ coincide with those studied previously by J. Lott.
For the entire collection see [Zbl 1192.00075].

##### MSC:
 19L10 Riemann-Roch theorems, Chern characters 58J28 Eta-invariants, Chern-Simons invariants 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 57R19 Algebraic topology on manifolds and differential topology
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