Manifolds with Fundamental Group a Generalized Free Product. I
نویسنده
چکیده
This announcement describes new methods in the study and classification of differentiable, PI or topological manifolds with infinite fundamental group. If, for example, Y + l is a closed manifold with n^ = Gx *H G2, n> 4, there is a decomposition Y~YtUx Y2 with ir1(Yx) = G1,irl(Yl) = G2, itt(X) = H. For large classes of fundamental groups, the codimension one splitting theorems of [C3], extending results of [Bl], [B2], [BL], [FH], [L], [W2], reduced the classification of manifolds homotopy equivalent to Y to the classifications of the manifolds homotopy equivalent to Yt and to Y2. However, the construction in [C4] and [C5] of manifolds V simple and tangentially homotopy equivalent to RP #RP9 k>09 but V is not a nontrivial connected sum, demonstrated in a strong way the existence of unsplittable manifolds and homotopy equivalences. In the present announcement we get around these difficulties by, roughly speaking, adapting methods of [C3] to construct an abelian group we call the unitary nilpotent group which depends on H, Glt G2 and which acts freely on the set of manifolds equipped with homotopy equivalences to F, with each coset of this action containing a unique split manifold. Thus, the classification of manifolds homotopy equivalent to Y is reduced to computing UNil .groups and to classifying split homotopy equivalences. In this setting, all earlier splitting theorems are reinterpreted as showing that for certain Hy Gl9 G2, the unitary nilpotent group vanishes. However, for H = 0, Z2 C Gl9 G2 ¥= 0, n = 4k and Y orientable, the corresponding unitary nilpotent group is not finitely generated [C4], [C6]. The unitary nilpotent groups are 2-primary and are defined algebraically in [C6] and depend only on the ring with involution Z[H], the Z[#]-bi-
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تاریخ انتشار 2007