In “Knots, sutures, and excision” (J. Differential Geom. 84, 301–364), Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.

Starting with a orientable compact real-analytic Riemannian manifold $(L,g)$ with $\chi (L) = 0$, we show that a small neighborhood $\mathrm{Op}(L)$ of the zero section in the cotangent bundle $T*L$ carries a Calabi–Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.

We establish a tight inequality relating the knot genus $g(K)$ and the surgery slope $p$ under the assumption that $p$-framed Dehn surgery along $K$ is an L-space that bounds a sharp $4$-manifold. This inequality applies in particular when the surgered manifold is a lens space or a connected sum thereof. Combined with work of Gordon–Luecke, Hoffman, and Matignon–Sayari, it follows that if surgery along a knot produces a connected sum of lens spaces, then the knot is either a torus knot or a cable thereof, confirming the cabling conjecture in this case.

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. We prove the following equivalences for asymptotically harmonic manifolds $X$ under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded: (a) $X$ has rank one; (b) $X$ has Anosov geodesic flow; (c) $X$ is Gromov hyperbolic; (d) $X$ has purely exponential volume growth with volume entropy equals $h$. This generalizes earlier results by G. Knieper for noncompact harmonic manifolds and by A. Zimmer for asymptotically harmonic manifolds admitting compact quotients.

We provide a necessary condition for the existence of a compact Clifford–Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi and Ono with the observation that a fiber bundle with contractible fiber induces an isomorphism between the cohomology rings of the total space and the base space. We give some examples: $SL(p + q,\mathbb{R})/SO(p, q)(p, q : \mathrm{odd})$, for instance—of homogeneous spaces that do not admit compact Clifford–Klein forms.

In this paper, we will use Kahn and Markovic’s immersed almost totally geodesic surfaces to construct certain immersed $\pi_1$-injective $2$-complexes in closed hyperbolic $3$-manifolds. Such $2$-complexes are locally almost totally geodesic except along a $1$-dimensional subcomplex. By using Agol’s result that the fundamental groups of closed hyperbolic $3$-manifolds are vitually compact special, and other works on geometric group theory, we will show that any closed hyperbolic $3$-manifold virtually contains any prescribed subgroup in the homological torsion. More precisely, our main result is, for any finite abelian group $A$, and any closed hyperbolic $3$-manifold $M, M$ admits a finite cover $N$, such that $A$ is a direct summand of $\mathit{Tor}(H_1(N; \mathbb{Z}))$.