Lower bounds for the degree of a branched covering of a manifold. (English. Russian original) Zbl 1402.57004

Gugnin, Dmitriĭ V.

Math. Notes 103, No. 2, 187-195 (2018); translation from Mat. Zametki 103, No. 2, 186-195 (2018).

This paper is about lower bounds for the number of sheets of a branched covering between two manifolds \(X\to Y\) of the same dimension. The main result is given in terms of the cohomology rings of the spaces involved. More precisely the author considers the following question: Suppose given two manifolds \(X^N\) and \(Y^N\) for which at least one branched covering \(f:X^N\to Y^N\) exists. What is a lower bound for the least degree \(n\) of such a branched covering in terms of the topology of these manifolds? The main result of the manuscript is:
Theorem 3: Let \(k\geq 1\) and \(N\geq 4k+2\) be any fixed integers. Given a connected closed orientable \(N\)-manifold \(X^N\) with \(L(X)=N\), let \[ Y^N:=S^2\times \cdots \times S^2\times S^{N-2k}. \] Then any \(n\)-fold branched covering \(f: X^N \to Y^{N-2k}\) satisfies the condition \(n\geq N-2k.\)
Here \(L(X)\) is the rational cohomological length of the space \(X\).
This improves substantially the best known general result for a lower bound for the integer \(n\). The main ingredient to prove the result is Lemma \(1\), which compares the ring structure of a ring obtained as the fixed subring of an action of a group \(G\) over a graded commutative algebra, with the ring obtained as the fixed subring of the restricted action to a subgroup \(H\subset G\) of finite index.

Reviewer: Daciberg Lima Gonçalves (São Paulo)

MSC:

57M12	Low-dimensional topology of special (e.g., branched) coverings
55M35	Finite groups of transformations in algebraic topology (including Smith theory)
20J06	Cohomology of groups

Keywords:

branched covering; degree; cohomological lenght; symmetric power space

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