Transfer, cap-product and Poincaré duality in homology and cohomology of real Lie groups (Q2791850)

Let \(\mathbf G\) be a real Lie group with a finite number of connected components, and let \(\Gamma\) be a co-compact subgroup of \(\mathbf G\) (that is, one such that \(\mathbf G/\Gamma\) is compact). The main purpose of this work is to define, in this context, operations of homological and cohomological restriction and transfer and to establish a transfer formula that relates them by use of a particular notion of cap product. The homology and cohomology in question here are those of the so-called \textit{differentiable \(\mathbf G\)-modules}, which are defined in Section 2.NEWLINENEWLINEA \textit{differentiable \(\mathbf G\)-module \(\mathbf M\)} is a (locally convex, separated and complete) topological vector space over \(\mathbb C\), together with a group homomorphism \(\mathbf G\rightarrow\mathrm{Aut}(\mathbf M)\) that satisfies some suitable conditions of differentiability and equicontinuity. For each such \(\mathbf M\), we define its related \textit{invariant} \(\mathbf M^{\mathbf G}=\{m\in\mathbf M:\forall g\in\mathbf G\), \(g\cdot m=m\}\) and \textit{coinvariant} \(\mathbf M_{\mathbf G}=\mathbf M/\langle\{g\cdot m-m:\forall g\in\mathbf G,\forall m\in\mathbf M\}\rangle\) \((\langle\dots\rangle\) denotes here the subgroup generated by the elements in the shown set). To get the homology \(H_\ast(\mathbf G,\mathbf M)\) of a differentiable \(\mathbf G\)-module \(\mathbf M\), one takes a projective resolution of \(\mathbf M\), suppressing the final term \(\mathbf M\), considers the complex obtained from it by taking the coinvariant part of each term, and calculates the homology of the resulting chain complex. To get its cohomology \(H^\ast(\mathbf G,\mathbf M)\), one takes an injective co-resolution (minus its first term), considers the invariant part of each term, and calculates the cohomology of the resulting chain complex. These definitions depend on the existence of such resolutions and co-resolutions for differentiable \(\mathbf G\)-modules, which is proven (by providing explicit examples) through the use of differential forms of compact support (for the projective resolution) and no restriction of support (for the injective resolution) in the work that culminates in Proposition 2.2.NEWLINENEWLINESection 3 starts by defining a version of cap product in this context (considering two differentiable \(\mathbf G\)-modules \(\mathbf M\) and \(\mathbf N\)). This is obtained again from some operations defined at the level of differential forms (Proposition 3.2) which, after applying the homology and cohomology constructions in the related complexes, provides the operation \(\cap:H_i (\mathbf G,\mathbf M)\times H^j(\mathbf G, \mathbf N)\rightarrow H_{i-j}(\mathbf G,\mathbf M\hat\otimes\mathbf N)\) (Here, \(\hat{\otimes}\) denotes the projective tensor product). Next, in Lemma 3.4, the author constructs isomorphisms between \(H_{s-i}(\mathbf G,\mathbf M)\) and \(H^i(\mathbf G,\mathbf M\otimes \delta_G)\) for each \(i\) and \(s\) and, having defined (after Lemma 3.6) a notion of fundamental class in this context, identifies these isomorphisms as versions of Poincaré duality (i.e., as being given by the cap product with the fundamental class). Here, \(\delta_G\) is the so-called \textit{module of the group \(\mathbf G\)}, which is is obtained from a left Haar measure \(dg\) on \(\mathbf G\) by applying a Radon-Nikodym construction.NEWLINENEWLINEIn Section 4, given \(\Gamma\) and \(\mathbf G\) as mentioned above, the transfer and restriction operations are defined. The homological and cohomological transfers are morphisms \(\mathrm{Tra}_G^\Gamma:H_i(\mathbf G,\mathbf M\otimes\delta_G^{-1})\rightarrow H_i(\Gamma,\mathbf M\otimes\delta_\Gamma^{-1})\) and \(\mathrm{Tra}^G_\Gamma:H^i(\Gamma,\mathbf M\otimes \delta_\Gamma)\rightarrow H^i(\mathbf G, \mathbf M\otimes \delta_G)\) that are induced from an integration over \(\mathbf G/\Gamma\) tensored by the modules of the groups \(\mathbf G\) and \(\Gamma\). The homological and cohomological restrictions \(\mathrm{Res}^G_\Gamma:H_i(\Gamma,\mathbf M)\rightarrow H_i(\mathbf G,\mathbf M)\) and \(\mathrm{Res}_G^\Gamma:H^i(\mathbf G,\mathbf M)\rightarrow H^i(\Gamma, \mathbf M)\) are induced resp. by the surjection \(\mathbf M_{\Gamma}\rightarrow\mathbf M_{\mathbf G}\) and the injection \(\mathbf M^{\mathbf G}\rightarrow\mathbf M^{\Gamma}\).NEWLINENEWLINEStill in Section 4, the transfer formula (for homology and cohomology) is Corollary 4.1 (of Proposition 4.4, establishing a similar formula for differential forms), and states that \(\mathrm{Res}_\Gamma^G(\mathrm{Tra}_G^\Gamma\cap\mathrm{Id})\) and \(\mathrm{Id}\cap\mathrm{Tra}^G_\Gamma\) are the same map \(H_i(\mathbf G,\mathbf M\otimes\delta_G^{-1}) \times H^j(\Gamma,\mathbf N\otimes\delta_\Gamma)\rightarrow H_{i-j}(\mathbf G,\mathbf M\hat{\otimes}\mathbf N)\).NEWLINENEWLINESection 5 relates the Poincaré duality isomorphism with the notions of restriction and transfer: Theorem 5.1 states that taking the cap product with the fundamental class turns the cohomology transfer homomorphism into the homology restriction homomorphism.NEWLINENEWLINEThe style of this work is very clear, and the proofs can be followed swiftly. For someone accustomed to the main definitions of differentiable \(\mathbf G\)-modules and their homology and cohomology, it should be enough to jump directly to sections 4 and 5, where the main new results are, going back to the earlier sections whenever necessary.