In this paper we introduce the following property, associated with connected, closed and smooth manifolds F: we say that F satisfies property CP (compatible with the point) if there exists a closed and smooth manifold M and a smooth involution T:M & RARR;M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:M \rightarrow M$$\end{document} such that its fixed point set is F boolean OR{point}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F \cup \{point\}$$\end{document}. The main motivation for such a definition is an old result obtained by combining theorems from the 50s and 60s, of John Milnor, Pierre Conner and Edwin Floyd, which says that the only spheres that satisfy such property are S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<^>1$$\end{document}, S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<^>2$$\end{document}, S4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<^>4$$\end{document} and S8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<^>8$$\end{document}. Later, in the 90s, P. Pergher solved this question for products of spheres, showing that relatively few (although infinite) such products satisfy CP. Basically, this paper is divided into two parts. In the first part, we get some simple general results that provide a reasonable amount of examples of validity and non-validity of this property; for example, all Cartesian product VxW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \times W$$\end{document}, where V and W are, respectively, m and n dimensional, with m+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m+n$$\end{document} odd and m and n having non disjoint dyadic expansions, does not satisfy CP. Also, all manifold of dimension 1, 2, 4 or 8 satisfies CP. In the (technically hard) second part of the paper we obtain some results of non-validity of property CP for Dold manifolds; these results illustrate the degree of difficulty in considering certain specific examples.
