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$ T\sp{3}$-actions on simply connected $ 6$-manifolds. I


Author: Dennis McGavran
Journal: Trans. Amer. Math. Soc. 220 (1976), 59-85
MSC: Primary 57E25
DOI: https://doi.org/10.1090/S0002-9947-1976-0415649-0
MathSciNet review: 0415649
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Abstract: We are concerned with $ {T^3}$-actions on simply connected 6-manifolds $ {M^6}$. As in the codimension two case, there exists, under certain restrictions, a cross-section. Unlike the codimension two case, the orbit space need not be a disk and there can be finite stability groups. C. T. C. Wall has determined (Invent. Math. 1 (1966), 355-374) a complete set of invariants for simply connected 6-manifolds with $ {H_\ast}({M^6})$ torsion-free and $ {\omega _2}({M^6}) = 0$. We establish sufficient conditions for these two requirements to be met when M is a $ {T^3}$-manifold. Using surgery and connected sums, we compute the invariants for manifolds satisfying these conditions. We then construct a $ {T^3}$-manifold $ {M^6}$ with invariants different than any well-known manifold. This involves comparing the trilinear forms (defined by Wall) for two different manifolds.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0415649-0
Keywords: Group actions, 6-manifold, torus, orbit space, cross-section, second Stiefel-Whitney class, surgery, connected sums, trilinear forms
Article copyright: © Copyright 1976 American Mathematical Society

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