Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, spectral projection Dirac operator finite trace in Atiyah von Neumann algebra. This allows us to define L2 index even case and its compatibility with Xie-Yu higher index. We also deduce versions classical Gromov-Lawson relative theorems. Finally, briefly di...

DISCLAIMER: This is just a draft. I am not sure of the final purpose of this document, so at the moment I have included lots of known results, in particular proofs of the usual Fundamental Theorem of Galois theory (in the finite and then the algebraic case). Seeing such proofs helps (me) to understand the analogies between the Fundamental Theorem presented here and the usual cases. Also, CAVEAT...

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p : (Q̃, q̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q) := Aut(p) = {g ∈ Adj(Q)′ | qg = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected c...

We prove the convergence of Bergman kernels and $$L^{2}$$ -Hodge numbers on a tower Galois coverings $$\{X_j\}$$ compact Kähler manifold X converging to an infinite (not necessarily universal) covering $${\widetilde{X}}$$ . also show that, as application, sections canonical line bundle $$K_{X_j}$$ for sufficiently large j give rise immersion into some projective space, if so do $$K_{{\widetilde...

Given an irreducible bivariate polynomial f (t, x) ∈ ℚ[t, x], what groups H appear as the Galois group of (t0, for infinitely many t0 ℚ? How often does a above f(t0, x), We give answer large x-degree with alternating or symmetric over ℚ(t). This is done by determining low genus subcovers coverings $$\tilde X \to \mathbb{P}_\mathbb{C}^1$$ monodromy groups.

In the context of a tower (strongly Birkhoff) Galois structures in sense categorical theory, we show that concept higher covering admits characterisation which is at same time absolute (with respect to base level tower), rather than inductively defined relative extensions lower order; and symmetric, depending on perspective terms arrows pointing certain chosen direction. This result applies the...