F. R. Cohen - On stunted projective spaces

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F. R. Cohen - On stunted projective spaces

F. R. COHEN, Department of Mathematics, University of Rochester, Rochester, New York 14627, USA

On stunted projective spaces

F. Sergeraert and V. Smirnov have recently studied the homology of the loop spaces for $RP^{\infty}/RP^n = X_n$ . It is the purpose of this note to record the homotopy type of these spaces after looping (sometimes more than once). The following is joint work with R. Levi.

Theorem. (1)   $\Omega(X_1)$ is homotopy equivalent to $S^1\times\Omega S^3$ .
(2)   $\Omega(X_2)$ is the homotopy theoretic fibre of a map $S^3\rightarrow Z$ where Z denotes the 6-skeleton of the Lie group G₂. Thus $\Omega^4_0(X_2)$ is homotopy equivalent to $\Omega^3_0S^3 \times \Omega^4_0(Z)$ .
(3)   $\Omega(X_3)$ is homotopy equivalent to $S^3 \times \Omega\bigl(S^7 \vee P^6(2)\bigr)$ .
(4)   $\Omega(X_n)$ is the homotopy theoretic fibre of a map from a finite complex to the Lie group Spin(n).

Further information concerning the map $X_n \rightarrow \Sigma{RP^n}$ is given. For example, the theorem above for X₂ gives a different proof of a theorem of Jie Wu concerning a splitting of the homotopy groups for $\Sigma RP^2$ .

Next: Gustavo Granja - On HPⁿ Up: 2) Homotopy Theory / Théorie Previous: Dan Christensen - Phantom