We show how to construct a Kirby diagram for a large class of finite volume hyperbolic 4-manifolds constructed by J. Ratcliffe and S. Tschantz. Date: October 16, 2018. 1 arXiv:1503.06722v1 [math.GT] 23 Mar 2015 2. Dualising the 24-cellIn this section we will explain to the reader how the 24-cell is constructed, and how we can visualise the boundary of the 24-cell in R 3 . We will not be going in to details about the structure of the 24-cell as our primary focus will be on handle decompositions. For more background information on the 24-cell we refer the reader to [1] chap.4. The paper [4] has some background on the 24-cell with some nice pictures depicting various edges and faces of the 24-cell.Let S ( * , * , * , * ) denote a sphere of radius 1 centred at a point in R 4 whose coordinates have two ±1's and whose other two coordinates are both zero. For example S (+1,+1,0,0) denotes the sphere of radius 1 centred at the point (1, 1, 0, 0) in R 4 . If we let H 4 denote the ball model of hyperbolic 4-space, then we find that all the spheres S ( * , * , * , * ) intersect the sphere at infinity orthogonally. This implies that each such sphere determines a hyperplane in H 4 . If we let Q ( * , * , * , * ) denote the corresponding half-space that contains the origin, and then take the intersection of all such half-spaces, we find that we obtain a 24-sided polyhedron P in H 4 . This polyhedron is known as the hyperbolic 24-cell and we will denote it by P . It is a four dimensional self dual polyhedron. We will denote the side of P that lies on the sphere S ( * , * , * , * ) also by S ( * , * , * , * ) . All the dihedral angles of P are π/2 and it has 24 vertices which are all ideal vertices. We can explicitly describe each ideal vertex: We have 8 vertices of the form v (±1,0,0,0) = (±1, 0, 0, 0), v (0,±1,0,0) = (0, ±1, 0, 0), v (0,0,±1,0) = (0, 0, ±1, 0), v (0,0,0,±) = (0, 0, 0, ±1) plus 16 vertices of the form v (±1/2,±1/2,±1/2,±1/2) = (±1/2, ±1/2, ±1/2, ±1/2). Finally let us mention that it has twenty four codimension 1 sides, ninety six codimension 2 sides, and ninety six codimension 3 sides.

We define functionals generalising the Seiberg-Witten functional on closed spin c manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then prove energy estimates along the flow, and establish local L 2 -derivative estimates. These are then used to show long time existence of the flow in sub-critical dimensions. In the critical dimension, we are able to show that long time existence is obstructed by an L k+2 curvature concentration phenomenon.Date: February 26, 2018.

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth.

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We consider perturbed quadharmonic operators, normalΔ4+V, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential V satisfying a bound from below by a non‐positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vector bundle and the underlying Riemannian manifold, we give a sufficient condition for the essential self‐adjointness of such operators. We then apply this to prove the separation property in L2 when the perturbed operator acts on functions.