A theorem on injectivity of the cup product

Wood, John C.

doi:10.1090/S0002-9939-1973-0307239-8

A theorem on injectivity of the cup product
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by John C. Wood

Proc. Amer. Math. Soc. 37 (1973), 301-304

DOI: https://doi.org/10.1090/S0002-9939-1973-0307239-8

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Abstract:

We prove that if a space X has abelian or sufficiently abelian fundamental group, then the cup product ${H^1}(X) \wedge {H^1}(X) \to {H^2}(X)$ is injective, giving an inequality between the associated Betti numbers. This generalises to a theorem of injectivity of the k-fold cup product on ${H^n}(X)$, given that the kth order Whitehead product on ${\pi _n}(X)$ is trivial or torsion.

References

Kuo-tsai Chen, A sufficient condition for nonabelianness of fundamental groups of differentiable manifolds, Proc. Amer. Math. Soc. 26 (1970), 196–198. MR 279822, DOI 10.1090/S0002-9939-1970-0279822-7
Heinz Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257–309 (German). MR 6510, DOI 10.1007/BF02565622
Kurt Reidemeister, Kommutative Fundamentalgruppen, Monatsh. Math. Phys. 43 (1936), no. 1, 20–28 (German). MR 1550506, DOI 10.1007/BF01707583
Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, DOI 10.1016/0040-9383(65)90039-X
G. J. Porter, Spaces with vanishing Whitehead products, Quart. J. Math. Oxford Ser. (2) 16 (1965), 77–84. MR 172292, DOI 10.1093/qmath/16.1.77