On the group of homotopy classes of relative homotopy automorphisms

Abstract

We prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

“…is finite by [18,Theorem 1.1]. The latter is a relative version of a Theorem of Sullivan [68, Theorem 10.2] which asserts that π 0 (hAut * (X)) → π 0 (hAut * (X Q )) has finite kernel whenever X is a finite and simply connected CW complex.…”

Mapping class groups of manifolds with boundary are of finite type

“…If X is 1-connected and of finite type, then π 0 hAutpX Q q is isomorphic to the Qpoints of a linear algebraic group (over Q, per Convention 2.2). Several constructions of such an isomorphism exist: Sullivan [Sul77] (see [BL05] for some additional details), Wilkerson [Wil76,Wil80], and Espic-Saleh [ES20] (strictly speaking, they require a choice of basepoint but since X is 1-connected pointed homotopy classes coincide with homotopy classes). A priori, these need not coincide though we expect they do; at least Sullivan's and Espic-Saleh's coincide [ES20,Theorem 4.11].…”

“…Several constructions of such an isomorphism exist: Sullivan [Sul77] (see [BL05] for some additional details), Wilkerson [Wil76,Wil80], and Espic-Saleh [ES20] (strictly speaking, they require a choice of basepoint but since X is 1-connected pointed homotopy classes coincide with homotopy classes). A priori, these need not coincide though we expect they do; at least Sullivan's and Espic-Saleh's coincide [ES20,Theorem 4.11]. In this paper we shall always consider this structure of a linear algebraic group on π 0 hAutpX Q q, which we will recall in Section 3.3.2.…”

“…The crucial property of minimal dg-Lie algebras is then that the group π 0 hAut dgLie pM q can be computed explicitly: π 0 hAut dgLie pM q " AutpM q{thomotopic to id M u, the quotient of the group AutpM q of automorphisms of M as a dg-Lie algebra, by the subgroup of automorphisms homotopic to the identity; this is implicit in [ES20]. The group AutpM q is easily seen to be isomorphic to the Q-points of a linear algebraic group, and under this isomorphism the subgroup of automorphisms homotopic to the identity is given by the Q-points of a unipotent closed subgroup.…”

“…(i) This paper crucially uses [ES20], which studies relative homotopy automorphisms. Apart from this, the arguments are standard with the exception of Corollary 2.14.…”