Book contents
- Frontmatter
- Preface
- Contents
- Chapter I BRAIDS AND CONFIGURATION SPACES
- Chapter II BRAIDS AND LINKS
- Chapter III POLYNOMIAL COVERING MAPS
- Chapter IV ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS
- Appendix 1 A presentation for the abstract coloured braid group (by Lars Gæde)
- Appendix 2 Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]
- Bibliography
- Index (to Chapters I–IV and Appendix 1)
Chapter IV - ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Preface
- Contents
- Chapter I BRAIDS AND CONFIGURATION SPACES
- Chapter II BRAIDS AND LINKS
- Chapter III POLYNOMIAL COVERING MAPS
- Chapter IV ALGEBRA AND TOPOLOGY OF WEIERSTRASS POLYNOMIALS
- Appendix 1 A presentation for the abstract coloured braid group (by Lars Gæde)
- Appendix 2 Threading knot diagrams (by Hugh R. Morton), [Math. Proc. Camb. Phil. Soc. 99(1986), 247–260]
- Bibliography
- Index (to Chapters I–IV and Appendix 1)
Summary
Throughout this chapter, X denotes a connected and locally pathwise connected topological space with the homotopy type of a CW–complex. Let C(X) denote the ring of complex valued, continuous functions on X. A Weierstrass polynomial P(x, z) over X can then be viewed as an element in the polynomial ring C(X)[z].
A natural first question concerning the algebra of a Weierstrass polynomial P(x, z) over X is to ask whether it has a root over C(X), in other words, whether there exists a continuous function λ : X → ℂ such that P(x, λ(x)) = 0. In §1, we shall present the basic elements of work of Gorin and Lin containing necessary and sufficient conditions that a space X has to satisfy in order that every simple Weierstrass polynomial P(x, z) of degree n ≥ 2 over X splits completely into linear factors over C(X). Associated with a simple Weierstrass polynomial P(x, z) over X we have the polynomial covering map T. E → X. Complete solvability of the equation P(x, z) = 0 is equivalent to triviality of π: E → X. This is an example of the connections between the algebra of the simple Weierstrass polynomial P(x, z) on the one hand and the topology of the polynomial covering map x: E → X on the other hand.
For an arbitrary covering map π. E → X, there is an induced monomorphism of rings π*: C(X) → C(E). Thereby we can consider C(E) as a ring extension of C(X), or as a C(X)–algebra.
- Type
- Chapter
- Information
- Braids and CoveringsSelected Topics, pp. 121 - 152Publisher: Cambridge University PressPrint publication year: 1989
- 1
- Cited by