Equivariant moduli problems, branched manifolds, and the Euler class
The authors give the following definitions. A \(G\)-moduli problem \((G\) is a compact oriented Lie group) is a triple (\textbf{B,E,S}) such that \({ \mathbf B}\) is a Hilbert manifold without boundary equipped with a smooth \(G\)-action; \textbf{E} is a Hilbert space bundle over \textbf{B}, also equipped with a smooth \(G\)-action, such that \(G\) acts by isometries on the fibres of \textbf{E} and the projection \({\mathbf E}\to{\mathbf B}\) is \(G\)-equivariant; \({\mathbf S}:{\mathbf B} \to{\mathbf E}\) is a smooth \(G\)-equivariant Fredholm section of constant Fredholm index such that the determinant bundle \(\det({\mathbf S})\to{\mathbf B}\) is oriented, \(G\) acts by orientation preserving isomorphisms on the determinant bundle, and the zero set \({\mathbf M}=\{x\in {\mathbf B}/{\mathbf S}(x)=0\}\) is compact. A finite dimensional \(G\)-moduli problem \((B,E,S)\) is called oriented if \(B\) and \(E\) are oriented and \(G\) acts on \(B\) and \(E\) by orientation preserving diffeomorphisms. A \(G\)-moduli problem (\textbf{B,E,S}) is called regular if the isotropy subgroup \(G_x=\{x\in G\mid g*x=x\}\) is finite for every \(x\in{\mathbf M}\). A morphism from a \(G\)-moduli problem (\textbf{B,E,S}) to another one \(\mathbf B',\mathbf E',\mathbf S'\)) is a pair \((\psi,\Psi)\) such that \(\psi\) is a smooth \(G\)-equivariant embedding of a neighbourhood \({\mathbf B}_0 \subset{\mathbf B}\) of \({\mathbf M}\) into \({\mathbf B} \), \(\Psi\) is a smooth injective bundle homomorphism of \({\mathbf E}_0={\mathbf E}_{{ \mathbf B}_0}\) into \({\mathbf E}'\) and a lift of \(\psi\), and the sections \textbf{S} and \textbf{S}' satisfy \({\mathbf S}'\circ\psi=\Psi\circ{\mathbf S}\), \({\mathbf M}'= \psi ({\mathbf M})\). Moreover, the linear operators \(d_x\psi:T_x{\mathbf B}\to T_{\psi (x)}{\mathbf B}'\) and \(\Psi_x:{\mathbf E}_x\to {\mathbf E}_{\psi(x)}'\) induce isomorphisms \(d_x\psi: \ker{\mathbf D}_x\to\ker {\mathbf D}_{\psi(x)}'\) and \(\Psi_x: \text{coker} {\mathbf D}_x \to\text{coker} D_{\psi(x)}'\) for \(x\in{\mathbf M}\), and the resulting isomorphism from \(\det({\mathbf S})\) to \(\det({\mathbf S}')\) is orientation preserving. Then, the authors establish the main result of the paper: ``There exists a functor, called the Euler class, which assigns to each compact oriented Lie group \(G\) and each regular \(G\)-moduli problem (\textbf{B,E,S}) a homomorphism \(\chi^{\mathbf{B,E,S}}:H_G^*({\mathbf B})\to \mathbb{R}\) \((H_G^*\) denotes equivariant cohomology with real coefficients) and satisfies the following: (Functoriality). If \((\psi,\Psi)\) is a morphism from (\textbf{B,E,S}) to \(({\mathbf B}',{\mathbf E}',{\mathbf S}')\) then \(\chi^{\mathbf{B,E,S}} (\psi^* \alpha)=\chi^{\mathbf{B',E',S'}}(\alpha)\) for every \(\alpha\in H_G^* ({\mathbf B}')\) (Thom class). If (B,E,S) is a finite dimensional oriented regular \(G\)-moduli problem and \((U,\tau)\) is a Thom structure on \((B,E,S)\) (i.e. \(U \subset E\) is a \(G\)-invariant open neighbourhood of the zero section that intersects each fibre in a convex set and \(U\cap E_K\) has compact closure for every compact subset \(K\subset B\); \(S^{-1}(U)\) has compact closure; \(\tau\in \Omega^n_G (E)\) (\(n=\text{rank} E)\) is an equivariant \(n\)-form such that \(d_G \tau=0\), \(\text{supp} (\tau)\subset U\), \(\int_{E_x} \tau=1\) for every \(x\in B)\), then \(\chi^{B,E,S}(\alpha) =\int_{B/G}\alpha \wedge S^*\tau\) for every \(\alpha \in H^*_G(B)\). Moreover, the Euler class is uniquely determined by these two properties (Functoriality) and (Thom class)''. The construction of the Euler class consists in to consider the (Thom class) axiom as the definition in the finite dimensional case (the existence of Thom structures is previously given) and they prove the (Functoriality) axiom in this context. Then, one reduces the infinite dimensional case to the finite dimensional case by using a real Hilbert space with a orthogonal linear \(G\)-action. Other properties of the Euler class (Transversality (If \textbf{S} is transversal to the zero section, then \(\chi^{ \mathbf{B,E,S}} (\alpha)=\int_{{\mathbf M}/G} \alpha)\)) Invariance by cobordism, Invariance by normal subgroups of \(G\) acting freely on \textbf{B}, and Rationality (If \(\alpha\in H^*_G({\mathbf B}; \mathbb{Q})\), then \(\chi^{\mathbf{B,E,S}}(\alpha)\in \mathbb{Q}))\) are established.
scientific article; zbMATH DE number 1929320