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We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordstrom, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstrom in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant.
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide
In this paper we describe the local limits under conjugation of all closed connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology.
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of $k$-differe
Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in cite{Nos} and with Reidemeister torsions.