论文标题
固定点集的平滑$ g $ -manifolds伪等于$ g $ -Template
Fixed point sets of smooth $G$-manifolds pseudo-equivalent to a $G$-template
论文作者
论文摘要
对于有限的组$ g $而不是主要电力订单的$ g $，奥利弗（Oliver，1996）回答了以下问题，哪些是在磁盘上$ g $的平滑动作的固定点集（euclidean Spaces）的固定点集。我们将Oliver的结果扩展到紧凑（分别，开放）平滑$ g $ -mmanifolds $ m $ pseudo-y $等效，是有限的$ \ m arthbb {z} $ - acyclic $ g $ -CW综合体$ n_g $是$ g $的Oliver号码。我们证明，上述问题的答案并不取决于$ y $的选择。 对于有限连接的$ g $ -CW复合$ y $，以使$ y^g $是非空的且连接的，称为$ g $ -Template，我们证明，紧凑型稳定的可行的可行流形$ f $是固定点集合的固定点集合$ m^g $紧凑型$ g $ g $ g $ - manifold $ mmanifold $ m $ m $ pseudo $ y $ $ y $ $ y $ y $ y $ if） χ（y^g）\ pmod {n_g} $。此外，存在与$ g $ -template $ y $相当的紧凑平滑固定点免费$ g $ -mmanifold pseudo-y $，而仅当$χ（y^g）\ equiv 0 \ pmod {n_g} $。特别地，与磁盘上的操作类似，存在一个紧凑的平滑固定点免费$ g $ -manifold伪伪伪型与真实的投射空间$ \ mathbb {rm} {\ rm p}^{\ rm p}^{2n} $，对于整数$ n \ geq 1 $，如果是$ g $，则是olive olive a olive op olive operive olive geq 1 $。最后，我们证明，每个有限的Oliver $ g $在$ \ mathbb {r} {\ rm p}^{2n} $上都有一个平滑的固定点免费操作，用于某些整数$ n \ geq 1 $。
For a finite group $G$ not of prime power order, Oliver (1996) has answered the question which manifolds occur as the fixed point sets of smooth actions of $G$ on disks (resp., Euclidean spaces). We extend Oliver's result to compact (resp., open) smooth $G$-manifolds $M$ pseudo-equivalent to $Y$, a finite $\mathbb{Z}$-acyclic $G$-CW complex such that the fixed point set $Y^G$ is non-empty, connected, and $χ(Y^G) \equiv 1 \pmod{n_G}$, where $n_G$ is the Oliver number of $G$. We prove that the answer to the question above does not depend on the choice of $Y$. For a finite connected $G$-CW complex $Y$ such that $Y^G$ is non-empty and connected, called a $G$-template, we prove that a compact stably parallelizable manifold $F$ occurs as the fixed point set $M^G$ of a compact smooth $G$-manifold $M$ pseudo-equivalent to $Y$, if and only if $χ(F) \equiv χ(Y^G) \pmod{n_G}$. Moreover, there exists a compact smooth fixed point free $G$-manifold pseudo-equivalent to a $G$-template $Y$, if and only if $χ(Y^G) \equiv 0 \pmod{n_G}$. In particular, similarly as for actions on disks, there exists a compact smooth fixed point free $G$-manifold pseudo-equivalent to the real projective space $\mathbb{R}{\rm P}^{2n}$ for an integer $n \geq 1$, if and only if $G$ is an Oliver group. Finally, we prove that each finite Oliver group $G$ has a smooth fixed point free action on $\mathbb{R}{\rm P}^{2n}$ itself for some integer $n \geq 1$.
