Given a 3–manifold the second author defined functions δn : H (M ;Z) → N, generalizing McMullen’s Alexander norm, which give lower bounds on the Thurston norm. We reformulate these invariants in terms of Reidemeister torsion over a non– commutative multivariable Laurent polynomial ring. This allows us to show that these functions are semi-norms.

The construction of Boehmians on amanifold requires a commutative convolution structure. We present such constructions in two specific cases: anN-dimensional torus and an N-dimensional sphere. Then we formulate conditions under which a construction of Boehmians on a manifold is possible.

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré du-ality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold.

If R is a commutative ring, M a compact R-oriented manifold and G a finite graph without loops or multiple edges, we consider the graph configuration space MG and a Bendersky–Gitler type spectral sequence converging to the homology H∗(M, R). We show that its E1 term is given by the graph cohomology complex CA(G) of the graded commutative algebra A = H∗(M, R) and its higher differentials are obt...

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists equivalently in the coordinate or the momentum planes embedded in the 4D c...

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists equivalently in the coordinate or the momentum planes embedded in the 4D c...

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists equivalently in the coordinate or the momentum planes embedded in the 4D c...

We define a notion of “Frobenius pair”, which is a mild generalization of the notion of “Frobenius object” in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a “commutative Frobenius pair” in the stable homotopy category of spectra.

We propose a path integral formulation of noncommutative generalizations of spacetime manifold in even dimensions, characterized by a length scale λP . The commutative case is obtained in the limit λP = 0. PACS numbers: 04.60.Gw, 04.60.-m, 02.40-k. ∗E-mail: [email protected]

A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions.