Free and semi-inert cell attachments

Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously studied inert condition. Under these conditions we determine $H_*(ΩY;R)$ as an $R$-module and as an $R$-algebra respectively. Under a further condition we show that $H_*(ΩY;R)$ is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.

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