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- W2022075286 abstract "According to Lpez de Medrano, a manifold with a free involution is determined by its Browder-Livesay index and the normal cobordism class of its orbit space. Here we study the Browder-Livesay indices of some cobordant involutions, namely, the characteristic submanifolds of a fixed involution. In [2] and [3], studying fixed point free involutions on homotopy spheres, W. Browder and G. R. Livesay defined an index a measuring the obstruction to finding an invariant embedded sphere of codimension 1. For a closed orientable manifold M4k-l with a fixed point free involution T which preserves orientation, the Browder-Livesay index o{T, A1) is defined as the signature of a bilinear form on the homology of a characteristic submanifold; a characteristic submanifold W is a closed, orientable submanifold of codimension 1 whose complement has two components A and B such that B = TA and Cl A n Cl B W. Since characteristic submanifolds arise as f-'RP'-' where f: M/T RP' (real projective space) is the classifying map of the Z2 bundle M M/T, characteristic submanifolds exist and any two are equivariantly cobordant in M x 1. It follows that or does not depend on choice of W [1]. However, if Mo and M, are characteristic submanifolds of (T, X4k), in general oMo0 v M1; if X is S4k, aMo oAM1l = 2o(T x 1; S x 1, MO M1) [1] and if MO and M1 are spheres, (T, MO) and (T, M1) are equivariantly diffeomorphic iff aMo = aM1 [2], [3]. Here we derive a formula for p(Mo, M1) = rM0 oAl1 2o(X xI,M0 -1) and show that it lies between ?dim H2kX and does assume all these values. We also give an algebraic condition for a subspace of H2kX to be H2kA where A n TA is a characteristic submanifold of X. Received by the editors August 27, 1973. AMS (MOS) subject classifications (1970). Primary 57E10." @default.
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- W2022075286 date "1975-02-01" @default.
- W2022075286 modified "2023-09-23" @default.
- W2022075286 title "Characteristic submanifolds of fixed point free involutions" @default.
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- W2022075286 doi "https://doi.org/10.1090/s0002-9939-1975-0388424-8" @default.
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