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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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