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Stable retraction of $$CW$$-complexes whose cohomologies are small exterior algebras. (English) Zbl 0811.55009
In [Q. J. Math., Oxf. II. Ser. 35, 115-119 (1984; Zbl 0555.55010)], F. R. Cohen and F. P. Peterson studied three cell complexes $$X= S^{a_ 1} \cup e^{a_ 2}\cup e^{a_ 1+ a_ 2}$$ satisfying the following conditions:
SR1 $$H^* (X,{\mathbf F}_ 2) \cong \Lambda (u_{a_ 1}, u_{a_ 2})$$ where $$\deg u_{a_ i}= a_ i$$ for $$i=1,2$$ and $$a_ 2> a_ 1$$,
SR2 $$Sq^{a_ 2- a_ 1} u_{a_ 1}= u_{a_ 2}$$,
SR3 $$Y= S^{a_ 1}\cup e^{a_ 2}$$ is a stable retract of $$X$$.
They proved that if such a space $$X$$ exists, then $$a_ 2- a_ 1=2^ t$$ for $$t=0$$, 1, 2, or 3 and $${{a_ 1}\choose {2^ t}} \equiv 1\bmod 2$$.
In this note we determine necessary and sufficient conditions for the existence of such a three cell complex in terms of the degrees of the generators of the cohomology. Theorem: There exists a CW-complex $$X$$ satisfying the above three conditions if and only if $$a_ 2- a_ 1= 2^ t$$ and $$a_ 1\equiv -1\bmod 2^{t+1}$$ for $$t=0$$, 1, 2, or 3. Corollary: $$E_ 6/ F_ 4$$ never retracts stably to its 17-skeleton. Thus $$E_ 6/ F_ 4$$ is not stably parallelizable.
##### MSC:
 55P99 Homotopy theory 55S10 Steenrod algebra 57T15 Homology and cohomology of homogeneous spaces of Lie groups
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