Chern–Simons theory, surface separability, and volumes of 3-manifolds

Abstract

We study the set vol(M, G) of volumes of all representations ρ : π1M → G, where M is a closed oriented 3-manifold and G is either Iso+H 3 or Isoe SL2(R).By various methods, including relations between the volume of representations and the Chern-Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover M with vol( M, Iso+H 3 ) = {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover M with vol( M, Isoe SL2(R)) = {0}.We also find 3-manifolds M with positive simplicial volume but vol(M, Iso+H 3 ) = {0}, and non-trivial graph manifolds M with vol(M, Isoe SL2(R)) = {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M, G) = {0}.Besides we determine vol(M, G) when M supports the Seifert geometry. ContentsDefinition 4.9. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let J 0 be a JSJ piece, T 0 be a JSJ torus adjacent to J 0 and ζ 0 be a slope on T 0 . A partial PW subsurface j : R M is said to be virtually bounded by ζ 0 outside J 0 if the boundary ∂R of R is non-empty, covering ζ 0 under j, and if the interiorR of R misses J 0 under j. In this case, the carrier chunk X(R) ⊂ M of R is the unique minimal chunk that contains R, and the carrier boundary of X(R) is the component T 0 ⊂ ∂X(R).Definition 4.10. We say that a partial PW subsurface j : R M is parallel cutting if, for every JSJ torus T ⊂ M , all components of j −1 (T ) in R cover the same slope of T . Virtual existence of partial PW subsurfacesTheorem 4.11. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let ζ 0 be a slope on a JSJ torus T 0 adjacent to a JSJ piece J 0 . Then, for some finite coverM of M together with an elevation (J 0 ,T 0 ,ζ 0 ) of the triple (J 0 , T 0 , ζ 0 ), Pierre Derbez I2M UMR 7373

A characterization of virtually embedded subsurfaces in $3$-manifolds

“…The construction for Corollary 1.4 relies on techniques developed by PW2,PW3], which can be used to produce closed or bounded virtually essentially embedded subsurfaces of 3-manifolds that intersect JSJ tori in certain controllable pattern. These so-called partial PW subsurfaces have been further investigated in [DLW,Section 4]. Understanding in details how such subsurfaces become virtually embedded leads to constructions crucial to the proof of Theorem 1.1.…”

Positive simplicial volume implies virtually positive Seifert volume for 3–manifolds

“…Theorem 3.1 (Additivity principle, [DLW,Theorem 3.5]; see also [DW2]). Let M be an oriented closed 3-manifold with JSJ tori T 1 , · · · , T r and JSJ pieces J 1 , · · · , J k , and let ζ 1 , · · · , ζ r be slopes on T 1 , · · · , T r , respectively.…”

“…According to [DLW,Proposition 4.2], there is a finite cover p : M → M such that each JSJ piece J of M that covers J factors through J ′ . In particular, in the notations we have just used, SV( J ( α)) > 0.…”

“…However, such information seems difficult to characterize. For example, the vanishing or non-vanishing of SV(N ) implies nothing about the behavior of HV(N ), [BG1,Section 4 and 5]; and except for the geometric case (Theorem 1.1 (2)), the geometry of pieces fails to detect the vanishing or non-vanishing of HV(N ) or SV(N ) either, [DLW,Theorem 1.7]. On the other hand, the existence of some finite cover of N with non-vanishing representation volume turns out to be a question more accessible.…”

Volume of representations and mapping degree

“…This invariant has been introduced and studied by R. Brooks and W. Goldman [BG1,BG2,Go] as a geometrical analogue of the celebrated simplicial volume of orientable closed manifolds due to M. Gromov [Gr,Th1]. During the past few years, much has been known about the ( SL 2 (R) × Z R)representation volume (the Seifert volume) and the PSL(2, C)-representation volume (the hyperbolic volume) for 3-manifolds and their finite covers [DW1,DW2,DLW,DSW,DLSW]. Those invariants have demonstrated to be useful in studying nonzero degree maps between 3-manifolds, especially when the simplicial volume vanishes.…”