Invariants preserved by mutation (Q384301)

It is known that certain types of generalized mutation of a manifold \(M\), i.e. cutting \(M\) along an embedded codimension 1 submanifold and regluing by a diffeomorphism \(\tau\) of the submanifold, preserves geometric invariants such as the volume. For example, by a result of \textit{D. Ruberman} [Invent. Math. 90, 189--215 (1987; Zbl 0634.57005)], classical mutation of a hyperbolic link in the 3-sphere preserves the volume of the link; \textit{W. D. Neumann} [Contemp. Math. 541, 233--246 (2011; Zbl 1237.57013)] observed that the \(PSL(2,\mathbb C)\)-fundamental class and the Bloch invariant of hyperbolic 3-manifolds are also preserved under mutation, and the present author gave a topological proof of Ruberman's theorem using the fundamental class construction. ``The aim of this paper is to prove in a general setting that \(G\)-fundamental classes and hence various geometric invariants are preserved under generalized mutation''. For a closed, orientable \(d\)-manifold \(M\) and a representation \(\rho: \pi_1M \to G\), one has the naturally associated \(G\)-fundamental class \((B\rho)_*[M] \in H_d(BG)\) where \(BG\) denotes the classifying space of some Lie group \(G\).NEWLINENEWLINEIn the present paper, a version of such a fundamental class is considered also for manifolds with boundary. Suppose that the interior of \(M\) is a \(\mathbb Q\)-rank 1 locally symmetric space of noncompact type, and that \(G\) is a semisimple Lie group without compact factor, where \(\rho\) sends the boundary components of \(M\) to some parabolic subgroup of \(G\). Then, if \(\rho^\tau: \pi_1M^\tau \to G\) is obtained by a generalized mutation of \(\rho\), the main result of the paper states that the suitably defined rational fundamental classes of \(\rho\) and \(\rho^\tau\) coincide (i.e., for homology with coefficients in \(\mathbb Q\); this does not remain true for integer coefficients). This implies then that generalized mutation preserves several geometric invariants of closed or \(\mathbb Q\)-rank 1 locally symmetric spaces such as volume, Goncharov invariant, Bloch invariants.