Dimensions of fixed point sets of involutions |
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Authors: | Pedro L Q Pergher Fábio G Figueira |
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Institution: | 1. Departamento de Matemática, Universidade Federal de S?o Carlos, Caixa Postal 676, S?o Carlos, SP, 13565-905, Brazil
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Abstract: | Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fn and F2 of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fn is small if the normal bundle of F2 does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that
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In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F2 in M; further, we determine in these cases the equivariant cobordism class of (M, T).
Received: 25 August 2005 |
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Keywords: | Primary 57R85 Secondary 57R75 |
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