In an appendix to an earlier paper we showed how to construct tunnels of positive scalar curvatur... more In an appendix to an earlier paper we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature.
Journal of the Indian Mathematical Society, Dec 1, 1976
Let X be a closed C°°-rrianifold of dimension N. Then two additional structures on X have been qu... more Let X be a closed C°°-rrianifold of dimension N. Then two additional structures on X have been quite extensively studied. One is the Riemannian structure giving rise to Riemannian geometry and the other one is the triangulation of X giving rise to polyhedral or combinatorial topology. Sometimes we come across a problem in which we have an invariant for the manifold which has a nice expression in terms of one of these structures and we want to express this invariant in terms of the other structure. Let us consider two examples. Our first example is related to Pontrjagin classes p i9 1 g i g N/4, of the manifold X. If we choose a Riemannian structure on X, then by the well-known Chern-Weil theory (see [2]) we can explicitly write down a closed 4/-form representing the ith Pontrjagin class regarded as an element of H 4, (Z, R). It is an open problem to obtain a combinatorial formula for Pontrjagin classes. We now come to our second example. Our second example refers to Reidemeister-Franz torsion. Let % be a representation of %\{X) by orthogonal n x n matrices, n a fixed positive integer. Let £ x be the flat vector bundle on X defined by this representation. Suppose that all the cohomology groups of X with coefficients in the vector bundle £ z are zero. Then one can define an invariant known as Reidemeister-Franz torsion (see [3] and [4]) depending on X and %. To define this torsion one first chooses a smooth triangulation of X and in terms of this triangulation one defines an invariant which one proves is not changed if we consider subdivisions of the triangulation. We do not have so far a definition for this torsion which does not make use of the choice of a triangulation. It is an interesting open problem to define this invariant in terms of a Riemannian structure. There is a conjecture of Ray and Singer (see [4]) in this direction. They define an invariant which they call analytic torsion. The Ray-Singer analytic torsion is
We give a characterization of Whitney forms on an n-simplex σ and prove that for every real value... more We give a characterization of Whitney forms on an n-simplex σ and prove that for every real valued simplicial k-cochain c on σ, the form Wc is the unique differential k-form ϕ on σ with affine coefficients that pulls back to a constant form of degree k on every k-face τ of σ and satisfies τ ϕ =< c, τ >.
Proceedings of the American Mathematical Society, Mar 1, 1979
The paper contains a vanishing theorem for L1 harmonic forms on complete rotationally symmetric R... more The paper contains a vanishing theorem for L1 harmonic forms on complete rotationally symmetric Riemannian manifolds. This theorem requires no assumptions on curvature.
Proceedings of the American Mathematical Society, 1994
We show that every compact smooth manifold of three or more dimensions carries a Riemannian metri... more We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.
In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approxima... more In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture in [DM]. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class.
We explore to what extent one may hope to preserve geometric properties of three dimensional mani... more We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.
In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approxima... more In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L 2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the "principle of not feeling the boundary" (due to M. Kac), for heat kernels associated to boundary value problems.
In an appendix to an earlier paper [1] we showed how to construct tunnels of positive scalar curv... more In an appendix to an earlier paper [1] we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature.
Des groupes magiques ou quand des sous-groupes libres affines opèrent proprement discontinûment s... more Des groupes magiques ou quand des sous-groupes libres affines opèrent proprement discontinûment sur R 3
We explore to what extent one may hope to preserve geometric properties of three dimensional mani... more We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.
In an appendix to an earlier paper we showed how to construct tunnels of positive scalar curvatur... more In an appendix to an earlier paper we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature.
Journal of the Indian Mathematical Society, Dec 1, 1976
Let X be a closed C°°-rrianifold of dimension N. Then two additional structures on X have been qu... more Let X be a closed C°°-rrianifold of dimension N. Then two additional structures on X have been quite extensively studied. One is the Riemannian structure giving rise to Riemannian geometry and the other one is the triangulation of X giving rise to polyhedral or combinatorial topology. Sometimes we come across a problem in which we have an invariant for the manifold which has a nice expression in terms of one of these structures and we want to express this invariant in terms of the other structure. Let us consider two examples. Our first example is related to Pontrjagin classes p i9 1 g i g N/4, of the manifold X. If we choose a Riemannian structure on X, then by the well-known Chern-Weil theory (see [2]) we can explicitly write down a closed 4/-form representing the ith Pontrjagin class regarded as an element of H 4, (Z, R). It is an open problem to obtain a combinatorial formula for Pontrjagin classes. We now come to our second example. Our second example refers to Reidemeister-Franz torsion. Let % be a representation of %\{X) by orthogonal n x n matrices, n a fixed positive integer. Let £ x be the flat vector bundle on X defined by this representation. Suppose that all the cohomology groups of X with coefficients in the vector bundle £ z are zero. Then one can define an invariant known as Reidemeister-Franz torsion (see [3] and [4]) depending on X and %. To define this torsion one first chooses a smooth triangulation of X and in terms of this triangulation one defines an invariant which one proves is not changed if we consider subdivisions of the triangulation. We do not have so far a definition for this torsion which does not make use of the choice of a triangulation. It is an interesting open problem to define this invariant in terms of a Riemannian structure. There is a conjecture of Ray and Singer (see [4]) in this direction. They define an invariant which they call analytic torsion. The Ray-Singer analytic torsion is
We give a characterization of Whitney forms on an n-simplex σ and prove that for every real value... more We give a characterization of Whitney forms on an n-simplex σ and prove that for every real valued simplicial k-cochain c on σ, the form Wc is the unique differential k-form ϕ on σ with affine coefficients that pulls back to a constant form of degree k on every k-face τ of σ and satisfies τ ϕ =< c, τ >.
Proceedings of the American Mathematical Society, Mar 1, 1979
The paper contains a vanishing theorem for L1 harmonic forms on complete rotationally symmetric R... more The paper contains a vanishing theorem for L1 harmonic forms on complete rotationally symmetric Riemannian manifolds. This theorem requires no assumptions on curvature.
Proceedings of the American Mathematical Society, 1994
We show that every compact smooth manifold of three or more dimensions carries a Riemannian metri... more We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.
In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approxima... more In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture in [DM]. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class.
We explore to what extent one may hope to preserve geometric properties of three dimensional mani... more We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.
In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approxima... more In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L 2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the "principle of not feeling the boundary" (due to M. Kac), for heat kernels associated to boundary value problems.
In an appendix to an earlier paper [1] we showed how to construct tunnels of positive scalar curv... more In an appendix to an earlier paper [1] we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a three dimensional manifold of constant sectional curvature. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature.
Des groupes magiques ou quand des sous-groupes libres affines opèrent proprement discontinûment s... more Des groupes magiques ou quand des sous-groupes libres affines opèrent proprement discontinûment sur R 3
We explore to what extent one may hope to preserve geometric properties of three dimensional mani... more We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.
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Papers by Jozef Dodziuk