If there is an order-preserving group isomorphism from a directed abelian group G with a finitely generated positive cone G+ onto a simplicial group, then G is called a semisimplicial group. By factoring out the torsion subgroup of a unital group having a finite unit interval, one obtains a semisimplicial unital group. We exhibit a representation for the base-normed space associated with a semi...

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated Kalgebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

In this paper we prove some results on combinatorics of triangulations of 3-dimensional pseudo-manifolds, improving on results of [NZ], and apply them to obtain a simplicial formula for the Chern-Simons invariant of an ideally triangulated hyperbolic 3-manifold. Combining this with [MN] gives a simplicial formula for the η invariant also. In effect, the main ingredient in the formula is the sum...

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

The notation and terminology used in this paper have been introduced in the following articles: [3], [19], [9], [10], [16], [20], [4], [5], [22], [23], [21], [1], [6], [17], [18], [2], [25], [26], [24], [15], [12], [13], [8], [14], and [7]. Let A be a non empty set, x be an element, and a be an element of A. Let us observe that (A 7−→ x)(a) reduces to x. Let A, B be non empty topological spaces...

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

We define natural A∞-transformations and construct A∞-category of A∞-functors. The notion of non-strict units in an A∞-category is introduced. The 2-category of (unital) A∞-categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Sta...

We show the existence of a subalgebra A ⊆ P(ω) that satisfies the following three conditions: • A is Borel (when P(ω) is identified with 2). • A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility). • The forcing notion (A,⊆) modulo the ideal FIN of finite sets collapses the continuum to א0.

Let G = GLn(q) be the general linear group of degree n ≥ 2 defined over a finite field Fq of characteristic p. We fix a prime l 6= p and let stand R for a local principal ideal domain having characteristic 0, maximal ideal lR, and containing a primitive p-th root of unity. Then the residue field K = R/lR has characteristic l and a primitive p-th root of unity. By a Steinberg lattice of G over R...