Nice and lazy cell attachments (Q1923537)

Let \(X\) be a 1-connected CW-complex of finite type and let \(f: S^n\to X\) be a continuous map, \(n\geq 2\). In this paper the authors consider the map \(H_*\Omega i:H_* (\Omega X; \mathbb{R}) \to H_*(\Omega Y; \mathbb{R})\), where \(i\) denotes the injection of \(X\) into \(Y=X \cup_f e^{n+1}\). The cell attachment \(f\) is called lazy if the connecting map \(\delta: \Omega Y \to F\) of the Puppe sequence \(\Omega Y \to F\to Y @>i>> Y\) induces the trivial map in homology, and is called nice if \(H_*i: H_*(\Omega X; \mathbb{R})/(f') \to H_* (\Omega Y; \mathbb{R})\) is an injection; here \(f'\) is the element of \(H_{n-1} (\Omega X; \mathbb{R})\) induced by \(f\). The authors define laziness algebraically and topologically. They give algebraic criteria for laziness and interesting examples of cell attachments.