Consider the following composite based map $$f: S^2 \xrightarrow{\sim} RP^2/RP^1 \to RP^\infty/RP^1$$ induced by the inclusion of the real projective plane $RP^2$ into infinite real projective space $RP^\infty$.

Consider ordinary homology with coefficients in the cyclic group ${\mathbb{Z}}/2$ of order two. Is the induced map $$(\Omega f)_*:H_q(\Omega S^2; {\mathbb{Z}}/2) \to H_q(\Omega (RP^\infty/RP^1);{\mathbb{Z}}/2)$$ a monomorphism for each $q\ge 0$?

  • $\begingroup$ I don't have time to work through the details, but I would start with the following fact: The fibration $\Omega S^2 \rightarrow PS^2\rightarrow S^2$ is a pull back of the fibration $\Omega \mathbb{R}P^\infty/\mathbb{R}P^1\rightarrow P\mathbb{R}P^\infty/\mathbb{R}P^1\rightarrow \mathbb{R}P^\infty/\mathbb{R}P^1$, then use naturality of spectral sequences and edge homomorphisms. I'm not sure if this approach will work - just a first thought. $\endgroup$ – Jason DeVito Sep 16 '14 at 13:56

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