The union of a connected manifold and a point as fixed set of commuting involutions

$(Z_{2})^{k}$-actions whose fixed data has a section

“…As we have seen in [5], the above result for k = 2 together with facts from [7] made it possible to obtain, up to bordism, all possible (Z 2 )…”

“…This method has shown itself particularly effective when A has a single element, because in this case we have seen that if (M, Φ) is a (Z 2 ) k -action fixing V n ∪ {p}, then its fixed data bears, in terms of bordism, a strong resemblance to the fixed data of σΓ k -actions from a given involution. As an illustration of this effectiveness we have proved further that (M, Φ) is really bordant to some action of type σΓ [7] and [8]). However, if A has more than one element the classification seems much more difficult, because in this case the results of [7] indicate the possibility of other classes added to those produced by the operations σΓ k t ; as we have seen in [5], this is exactly what happens when V n = RP (2r) (Royster has proved in [2] that in this case A has more than one element), and in this case the method of [7] is not enough to determine the classification.…”

“…In this paper we will complete this classification for any k, by putting together in a similar way the above general case and facts from [7]. Referring to this classification, we remark that in [7] we developed a method to analyse the bordism classes of (Z 2 ) k -actions fixing the disjoint union of a connected closed n-dimensional manifold V n and a point p, having as a starting point the knowledge of the set A of all equivariant bordism classes of involutions fixing V n ∪ {p}.…”

“…In this paper we will complete this classification for any k, by putting together in a similar way the above general case and facts from [7]. Referring to this classification, we remark that in [7] we developed a method to analyse the bordism classes of (Z 2 ) k -actions fixing the disjoint union of a connected closed n-dimensional manifold V n and a point p, having as a starting point the knowledge of the set A of all equivariant bordism classes of involutions fixing V n ∪ {p}. This method has shown itself particularly effective when A has a single element, because in this case we have seen that if (M, Φ) is a (Z 2 ) k -action fixing V n ∪ {p}, then its fixed data bears, in terms of bordism, a strong resemblance to the fixed data of σΓ k -actions from a given involution.…”

“…However, if A has more than one element the classification seems much more difficult, because in this case the results of [7] indicate the possibility of other classes added to those produced by the operations σΓ k t ; as we have seen in [5], this is exactly what happens when V n = RP (2r) (Royster has proved in [2] that in this case A has more than one element), and in this case the method of [7] is not enough to determine the classification. Our main theorem thus constitutes an additional tool to handle this question, and the above classification for any k is an interesting example illustrating both the need for our theorem and the generality of the results of [7].…”