Advances in Geometry
Managing Editor: Grundhöfer, Theo / Joswig, Michael
Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard
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For a manifold M, the structure set S (M, rel ∂ ) is the collection of manifolds homotopy equivalent to M relative to the boundary. Siebenmann [R. C. Kirby, L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations. Princeton Univ. Press 1977] showed that in the topological category, the structure set is 4-periodic: S (M, rel ∂ ) ≅ S (MｘD 4 , rel ∂ ) up to a copy of ℤ. The periodicity has been extended to topological manifolds with homotopically stratiﬁed group actions for various representations in place of D 4, including twice any complex representation of a compact abelian group. In this paper, we extend the result to twice any complex representation of a compact Lie group. We also prove the bundle version of the periodicity.
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