Ações de Z2k com o conjunto de pontos fixos conexo e a propriedade CP

Abstract

It is well known that if $\phi:G \times M^m \rightarrow M^m$ is a smooth action of a compact Lie group on a closed smooth manifold, then its fixed point set $F_{\phi} = \bigcup\limits_{i=0}^{n}F^{i}$ is a disjoint union of closed submanifolds of $M^m$, where $F^i$ denotes the union of $i$-dimensional components of $F_{\phi}$. In this way, given a compact Lie group $G$ and a union of closed smooth manifolds $F = \bigcup\limits_{i=0}^{n}F^{i}$, we can ask ourselves if exists such a $G$-action defined on a closed smooth manifold $M^m$ whose fixed point set is $F$. In this work, we discuss this problem for the cases where $G=\mathbb{Z}_2$ with $F=P(m,n) \cup \{point\}$, here $P(m,n)$ denoting a Dold manifold (see \ref{ExemploDold}) and $\{point\}$ is a unique point; and $G=\mathbb{Z}_2^k$ with $F=F^n$ or $F=F^n \cup F^{n-1}$ with $F^n$ and $F^{n-1}$ being connected. Here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting involutions $T_1, \ldots, T_k$. Another question is concerning the CP property: more specifically, we say that a closed smooth manifold $F^n$ satisfies the CP property (compatible with the point) if there exists an involution $T:M^m \rightarrow M^m$ whose fixed point set is $F^n \cup \{point\}$. In chapter 3 we prove that the Dold manifolds $P(2^t-2,1)$ and $P(2,2^s-1)$ satisfy the CP property for all $t,s>1$. On the other hand, we will see that $P(m,n)$ do not satisfies CP for certain values of $m$ and $n$ (see the Introduction for details). This property was introduced in \cite{TeseJessica}, where several correlated results were obtained. In \cite{StongKos} Stong and Kosniowisky showed that if the fixed point set of an involution $(M^m,T)$ has only $n$-dimensional components and $m>2n$, then $(M^m,T)$ bounds equivariantly. In the same work, they proved that, if $m=2n$, then $(M^m,T)$ is equivariantly cobordant to the twist involution $(F^n \times F^n, \tau)$ where $\tau(x,y) = (y,x)$. In \cite{Onzedoiska} Pergher extended this result for $\mathbb{Z}_2^k$-actions $(M^m,\phi)$ whose fixed point set $F^n$ is connected. More specifically, he showed that, under these fixed point set conditions, if $m > 2^k n$, then $(M^m,\phi)$ bounds equivariantly, and if $m=2^k n$, then $(M^m,\phi)$ is equivariantly cobordant to the $\mathbb{Z}_2^k$-twist (see definition \ref{Z2ktwist}). In \cite{stongclass} Stong realized the classification of all cobordism classes of involutions $(M^m,T)$ whose fixed point set has only n-dimensional components and $m=2n-1$. In \cite{zedoisdois} Pergher extended this result for $\mathbb{Z}_2^2$-actions $(M^m,\phi)$ whose fixed point set $F^n$ is connected, with $m=4n-1$ and $m=4n-2$. In chapter 4 we extend this work of Pergher for $\mathbb{Z}_2^k$-actions by determining all possible cobordism class of $\mathbb{Z}_2^k$-actions whose fixed point set $F^n$ is connected and $2^k n-2^{k-1} \leq m < 2^k n$. The fixed-data of a $\mathbb{Z}_2^k$-action $(M^m,\phi)$ fixing $F$, denoted by $(F,\{\xi_{\rho}\})$, is $F$ with a list of $2^k-1$ vector bundles over $F$, where the vector bundles $\xi_{\rho}$ are obtained by a decomposition of the normal bundle of $F$ in $M^m$. In \cite{zedoisdois2}, P. Pergher and F. Figueira showed the following result: let $(M^m,\phi)$ be a $\mathbb{Z}_2^2$-action with fixed-data $(F^n;\xi_{\rho_1},\xi_{\rho_2},\xi_{\rho_3}) \cup (F^{n-1};\mu_{\rho_1},\mu_{\rho_2},\mu_{\rho_3})$, and suppose that there are at least two vector bundles in $\{\xi_{\rho_1},\xi_{\rho_2},\xi_{\rho_3}\}$ that have dimension greater than $n$, and at least one $\mu_{\rho}$ has dimension greater than $n-1$. Then $(M^m,\phi)$ bounds equivariantly. In that paper, the authors proposed the following generalization for $\mathbb{Z}_2^k$-actions: \textbf{Conjecture:} Let $(M,\psi)$ be a smooth $\mathbb{Z}_2^k$-action with fixed-data $(F^n, \{ \xi_{\rho} \}_{\rho}) \cup (F^{n-1}, \{ \mu_{\rho} \}_{\rho})$. Suppose that at least $2^{k-1}$ $\xi_{\rho's}$ over $F^n$ have dimension greater than $n$ and at least one $\mu_{\rho}$ has dimension greater than $n-1$. Then $(M,\psi)$ bounds equivariantly. This is the main result of chapter 5 of this thesis. But we achieved the following improvement of the above conjecture: the condition that at least one $\mu_{\rho}$ has dimension greater than $n-1$ can be removed.