loop theorem

《Mathematical Physics: Classical Mechanics》 · 第 112 页 · Chapter 11，Theorem 11.1（完整陈述）

Let \(M\) be a (possibly non-compact) 3-manifold with boundary \(\partial M\) such that the inclusion-induced homomorphism \(\Pi_1(\partial M)\longrightarrow\Pi_1(M)\) is not injective. Then there exists a (piecewise linear) embedding of the disc \(e:D^2\longrightarrow M\), with \(e^{-1}(\partial M)=\partial D^2\) such that the restriction \(e:\partial D^2\longrightarrow\partial M\) is not homotopic to a constant map.