Topology of the moduli of representations with Borel mold (Q1884538)

A representation with Borel mold of a monoid \(\Gamma\) on a scheme \(X\) is a monoid homomorphism \(\Gamma \to M_n\) \((\Gamma (X, \theta_x))\) which is locally equivalent to one having its image in and generating the subalgebra of upper triangular matrices. A general construction has been given by the first author for the moduli of such representations. In the present paper the special case of the free monoid is studied. An explicit construction is given for the moduli \(Ch_n(m)_B\) for the free monoid of rank \(m\) as a fibre bundle over the configuration space \(F_n({\mathbb A}_{\mathbb Z}^m)\) of the affine space \({\mathbb A}_{\mathbb Z}^m\) with fibre \(({\mathbb P}_{\mathbb Z}^{m-2})^{n-1} \times ({\mathbb A}_{\mathbb Z}^{m-1})^{(n-2)(n-1)/2}\). The Serre spectral sequence is then used to compute the cohomology ring \[ H^*(Ch_n(m)_B)\cong H^*(F_n ({\mathbb C}^m)) \otimes {\mathbb Z} [t_1, \dots, t_{n-1}]/(t_1^{m-1}, \dots, t_{n-1}^{m-1}). \] Moreover, the associated virtual Hodge polynomial is calculated. In a final section the case of infinite rank monoids is considered.