A converse to (Milnor-Kervaire theorem) ×Retc…

A Converse to Dye’s Theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Ribet’s Converse to Herbrand’s Theorem

Weil Converse Theorem

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

A Torsion - Free Milnor - Moore Theorem

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let R ⊂ Q be a subring containing 1/2. Let ρ(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M/TorsionM . We show that the inclusion P ⊂ FH∗(ΩX ;R) of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras UP ∼ = −→ FH∗(ΩX ;R), provi...

A Torsion - Free Milnor - Moore Theorem Jonathan

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let p be an odd prime. We prove the theorem: If p ≥ n/q, then the Hurewicz homomorphism induces isomorphisms U(Fπ∗(ΩX(p))) = −→ FH∗(ΩX(p)) and UE ∞ π (ΩX) = −→ E∞ H (ΩX) of Hopf algebras, where E ∞ π and E ∞ H are the limit terms of the homotopy and homology Bockstein spectral sequences, respectively, a...