Groups, coverings and Galois theory. (English) Zbl 0749.57001
For a space \(X\), the ring of complex-valued continuous functions on \(X\) is denoted by \(C(X)\). Any \(n\)-fold cover \(\pi: E\to X\) induces a monomorphism \(\pi^*: C(X)\to C(E)\) so that \(C(E)\) is an algebra over \(C(X)\). Let \({\mathcal G}\) be the group of automorphisms of \(C(E)\) over \(C(X)\), i.e. of ring homomorphisms \(\phi: C(E)\to C(E)\) with \(\phi\pi^*=\pi^*\). Say that \(C(E)\) is a normal extension of \(C(X)\) provided that \(\{f\in C(E)/\forall\phi\in {\mathcal G},\quad \phi(f)=f\}=C(X)\). The paper develops a Galois theory relating subgroups of \(\mathcal G\) to intermediate covers of \(X\). In particular, if \(H\) is a finite index subgroup of \(\pi_ 1(X)\) and \(\pi: E\to X\) the corresponding cover, then \(H\) is a normal subgroup if and only if \(C(E)\) is a normal extension of \(C(X)\).
After further development of the theory it is applied to covers associated with Weierstraß polynomials over \(X\), i.e. maps \(P: X\times \mathbb{C}\to \mathbb{C}\) with \(P(x,z)=z^ n+\sum^ n_{i=1}a_ i(x)z^{n-i}\), where \(a_ i: X\to \mathbb{C}\) is continuous. If for each \(x\), \(P(x,z)\) has no multiple roots, then it gives rise to an \(n\)-fold cover \(\{(x,z)\in X\times\mathbb{C}/P(x,z)=0\}\to X\) by projection. Such polynomials split uniquely into irreducible factors whose Galois group action classifies the associated cover.
After further development of the theory it is applied to covers associated with Weierstraß polynomials over \(X\), i.e. maps \(P: X\times \mathbb{C}\to \mathbb{C}\) with \(P(x,z)=z^ n+\sum^ n_{i=1}a_ i(x)z^{n-i}\), where \(a_ i: X\to \mathbb{C}\) is continuous. If for each \(x\), \(P(x,z)\) has no multiple roots, then it gives rise to an \(n\)-fold cover \(\{(x,z)\in X\times\mathbb{C}/P(x,z)=0\}\to X\) by projection. Such polynomials split uniquely into irreducible factors whose Galois group action classifies the associated cover.
Reviewer: D.B.Gauld (Auckland)
MSC:
57M12 | Low-dimensional topology of special (e.g., branched) coverings |
13B25 | Polynomials over commutative rings |
46J10 | Banach algebras of continuous functions, function algebras |
57M10 | Covering spaces and low-dimensional topology |