Loop homological invariants associated to real projective spaces

Let A be a based subspace of Y. Under the assumptions that Y is path-connected and that the reduced diagonal map of A induces the zero map in all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced Poincaré series of the loop space $Ω((A \wedge \mathbb{RP}^\infty) \cup_{A \wedge \mathbb{RP}^1} (Y \wedge \mathbb{RP}^1))$. Here $\mathbb{RP}^\infty$ and $\mathbb{RP}^1$ denote the infinite real projective space and the real projective line respectively.

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