Deficiences of immersions (Q1058760)

Let X and Y be manifolds of the same dimension \(n\geq 2\) and let \(f: X\to Y\) be an immersion with \(p=\sup \{n(y):\) \(y\in Y\}<\infty\) where \(n(y)=cardinality f^{-1}(y)\). If Y is compact and X is not, then \(n(y)<p\) for some \(y\in Y\). If Y is compact and simply connected and \(p\geq 2\), then Y contains a compact set E such that Y-E is not simply connected and n(y)\(\leq p-2\) for all \(y\in E\).