Quadratic functions in geometry, topology, and \(M\)-theory
This paper provides a theory of quadratic functions that appear in global analysis (e.g., the Riemann parity) and algebraic topology (various quadratic refinements of intersection pairing) and is particularly concerned with the relationship between analytically and topologically defined quadratic functions. Here is a very brief review of some ideas and results of the paper. The review is necessarily far from being complete and misses many lovely ideas of the authors. The authors start with some examples of the (well understood) relationship between analytically and topologically defined quadratic functions that appear in the theory of Riemann surfaces and 4-dimensional manifolds, indicate obstructions to understand an analogous relationship in higher dimensions, and then provide a mathematical apparatus that enables them to formulate such a generalization. The first of the examples is provided by the index theorem for Spin\(^{c}\)-Dirac-type operators on a Spin\(^{c}\)-4-manifold \(M\) (i.e., a manifold the second Stiefel-Whitney class of which is the mod 2 reduction of an integral cohomology class) and a relationship between the index and intersection pairing. Thus let us recall that Spin\(^{c}\)-structures (if any) are in a 1-1 correspondence with \(H^{2}(M;\mathbb{Z})\), Spin\(^{c}\)-structures correspond to naturally defined complex line bundles (which are also classified by \(H^{2}(M;\mathbb{Z})\)) and an elliptic Dirac-type operator. Then, given a Spin\(^{c}\)-structure, the line bundle corresponding to this structure, and the first Chern class \(\lambda \) of the line bundle, the index theorem gives the following formula for the index \(\kappa (\lambda )\) (this notation is derived from surgery theory, in particular the theory of Kervaire-Arf invariants) of the corresponding Spin\(^{c}\)-operator: \(\kappa (\lambda )=\frac{1}{8}\int_{M}(\lambda ^{2}-L_{4k})\), where \(L_{4k}\) is the \(4k\)-th component of the Hirzebruch polynomial (and \(k=1\) in this instance). Now, the Chern class \(\lambda \) is well known to be the characteristic element for the intersection pairing on \(H^{2}(M;\mathbb{Z})\), i.e., \(\int_{M}x\cup x=\int_{M}x\cup \lambda \) mod\,2; in other words, \(\lambda \) is a lift to \(H^{2}(M;\mathbb{Z})\) of the Wu class \(\nu _{2}\). Then the quadratic function \(q(x)=\frac{1}{2}\int_{M}(x^{2}-x\lambda )\), which is a refinement of the intersection pairing, gives the change of the index \(\kappa (\lambda )\) corresponding to the change \(\lambda \rightarrow \lambda -2x\) of the Spin\(^{c}\)-structure. Integrality of \(\kappa (\lambda )\) has also a purely algebraic explanation: The square of \(\lambda \) equals the signature mod 8 (by standard theorems of even unimodular quadratic forms). The authors indicate that much of this can be repeated in higher dimensions. For an integer lift \(\lambda \) of the Wu class \(\nu _{2k}\) the number \(\kappa (\lambda )=\frac{1}{8}\int_{M}(\lambda ^{2}-L_{4k })\) is also an integer and \(q(x)=\kappa (\lambda -2x)-\kappa (\lambda )\) is again a quadratic refinement of the intersection pairing. Moreover, (which is a non-trivial result proved in a separate Appendix) a Spin\(^{c}\)-structure on a \(4k\)-dimensional manifold determines an integral Wu structure and \(\kappa (\lambda )\) is the index of an elliptic operator. What remains to be clarified in higher dimensions is the behavior of \(\kappa (\lambda )\) under the variation \(\lambda \rightarrow \lambda -2x\). This dependence is clear in dimension 4: the variation \(\lambda \rightarrow \lambda -2x\det \)er\(\min \)es the variation of the Dirac operator, but clearly there is no such simple dependence in higher dimensions, and explaining this dependance in higher dimensions seems to be one of the main goals of the paper. Another inspiration comes from the equality of the (analytically defined) Riemann parity \(q(x)\) of the theta characteristic \(x\) on a Riemannian surface (recall that \(x\) is the square root of the line bundle of holomorphic 1-forms on \(M\) and \(q(x)\) is the dimension mod 2 of the space of holomorphic sections of \(x\)) and the (topologically defined) Pontryagin obstruction for a map \(f:S^{n+2}\rightarrow S^{n}\) to be null-homotopic (recall that the Pontryagin invariant, which lies in \(\pi _{1}(SO)=\mathbb{Z}/2\), is the surgery obstruction to kill the first homology group of the (framed) Riemannian surface \(f^{-1}(x)\) for a regular value \(x\in S^{n}\) by framed surgery.) The equality of these two invariants was discovered by Atiyah who identified theta characteristics with Spin structures and \(q(x)\) with the index of the Dirac operator mod 2, thus proving Spin-cobordism invariance of \(q(x)\). In dimension 2 the Spin cobordism is the same as the framed cobordism and both the Riemann and Pontryagin invariants give the isomorphism of this group with \(\mathbb{Z}/2\), which gives the desired equality. The authors give another proof of this equality (as well as independence of \(q(x)\) under holomorphic deformations). They consider holomorphic families \(E/S\) of Riemannian surfaces and determinant line bundles \(\det L\) of line bundles \(L\) over \(E\) with the property that \(L^{2}\) is isomorphic to the bundle of holomorphic 1-forms along the fibres of \(E/S\) and prove that for \(s\in S\) \(q(x)=(-1)^{\dim H^{0}(L_{s})}\) (\(L_{s}\) is the fibre of \(L\) over \(s\)). In this context a formula (roughly, index theorem for families) similar to that for the index of a Spin\(^{c}\)-operator over a 4-manifold arises: \(c_{1}(\det L)=\frac{1}{2}\int_{E/S}(x^{2}-xc_{1})\), where \(x\) is the Chern class of \(L\) and \(c_{1}\) is the first Chern class of the relative tangent bundle of \(E/S\). Thus the authors look for a category of objects classified by higher integral cohomology groups in a fashion analogous to that the group \(H^{2}(\cdot ,Z)\) classifies complex line bundles. The objects turn out to be differential cocycles \(\widehat{H}(M)\) (much in the style of Cheeger-Simmons characters; two dimensional differential cocycles classify line bundles with a fixed connection), and the bulk of the paper is devoted to the study of differential cocycles and their topological counterpart, which is the generalized differential cohomology theory (a refinement of ordinary generalized cohomology theory represented by an \(\Omega \)-spectrum formulated in the language of differential functions (a notion introduced by the authors) and simplicial sets; in the case of the differential refinement of the Eilenberg-McLane spectrum the relationship is analogous to that between ordinary cohomology theory and maps to the ordinary Eilenberg-McLane spectrum). Using differential cocycles the authors prove the following generalization of the index theorem, which is one of the main results of the paper. Let \(E/S\) be a map of manifolds of relative dimension \(4k-i\), \(i\leq 2\) (which is \(\widehat{H}\)-oriented). Let \(\nu \in Z^{2k}(E;\mathbb{Z}/2)\) be a cocycle representing the Wu class \(\nu _{2k}\), and let \(\widehat{H}_{\nu }^{2k}(E)\subset \widehat{H}^{2k}(E)\) be the subcategory of differential cocycles \((c,h,\varpi )\in C^{2k}(E;Z)\times C^{2k-1}(E;R)\times \Omega ^{2k}(E)\) such that \(c=\nu \bmod \mathbb{Z}\). Theorem. There exists a functor \(\kappa :\widehat{H}_{\nu }^{2k}(E)\rightarrow \widehat{H}^{i}(S)\) with the following properties: 1. \(\kappa (\lambda )=\frac{1}{8}\int_{E/S}(\lambda ^{2}-L_{4k})\) modulo torsion (one thinks about \(\lambda \) as an analogue of the canonical bundle); 2. For a fixed \(\lambda \) the functor \(q^{\lambda }:\widehat{H}^{2k}(E)\rightarrow \widehat{H}^{i}(S)\), defined by \(q^{\lambda }(x)=\kappa (\lambda -2x)\), is a quadratic refinement of the pairing \(B(x,y)=\int_{E/S}x\cup y\); 3. Both the functors have additional good properties (symmetry, transitivity, good behavior under basis change, etc.).
differential cocycles, differential function spaces, differential cohomology theories
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