In both ordinary and equivariant 3-dimensional topology there are strong uniqueness theorems for connected sum decompositions of manifolds, but in ordinary higher dimensional topology such decompositions need not be unique. This paper constructs families of manifolds with smooth group actions that are equivariantly almost sentations for which one summand is fixed. The examples imply the need for restrictions in any attempt to define Atiyah-Singer type invariants for odd dimensional manifolds with nonfree smooth group actions. Applications to other questions are also considered.
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