Combinatorial enumeration of cyclic covers of P1

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

Download Full-text

A remark on branched cyclic covers

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(93)90111-6 ◽

1993 ◽

Vol 87 (3) ◽

pp. 237-240

Author(s):

Jonathan A. Hillman

Keyword(s):

Cyclic Covers ◽

Branched Cyclic Covers

Download Full-text

An Introductory Set of Activities Designed to Facilitate Successful Combinatorial Enumeration for Undergraduate Students

PRIMUS ◽

10.1080/10511970.2016.1194934 ◽

2016 ◽

Vol 26 (10) ◽

pp. 889-904

Author(s):

Elise Lockwood ◽

Craig A. Swinyard

Keyword(s):

Undergraduate Students ◽

Combinatorial Enumeration

Download Full-text

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006580 ◽

2008 ◽

Vol 17 (10) ◽

pp. 1199-1221 ◽

Cited By ~ 6

Author(s):

TERUHISA KADOKAMI ◽

YASUSHI MIZUSAWA

Keyword(s):

Number Field ◽

Class Number ◽

Homology Sphere ◽

Rational Homology ◽

Homology Groups ◽

Cyclic Covers ◽

Class Number Formula ◽

Number Formula ◽

Iwasawa Invariants ◽

Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

Download Full-text