Real algebraic actions on projective spaces - A survey

Petrie, Ted. "Real algebraic actions on projective spaces - A survey." Annales de l'institut Fourier 23.2 (1973): 135-150. <http://eudml.org/doc/74119>.

@article{Petrie1973,
abstract = {Let $G$ be a compact lie group. We introduce the set $\{\bf S\}_G(Y)$ for every smooth $G$ manifold $Y$. It consists of equivalence classes of pair $(X,f)$ where $f:X\rightarrow Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$. Two pairs $(X_i,f_i)$, for $i=0,1$, are equivalent if there is a $G$ homotopy equivalence $\varphi :X_0\rightarrow X_1$ such that $f_0$ is $G$ homotopic to $f_1\circ \varphi $.Properties of the set $\{\bf S\}_G(Y)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G=S^1$ and $Y$ is the $S^1$ manifold defined by a “linear” $S^1$ action on complex projective $n$ space $\{\bf C\}\{\bf P\}^n$, we exhibit non-trivial elements of $\{\bf S\}_\{S^4\}(Y)$ by discussing a real algebraic action of $S^1$ on $\{\bf C\}\{\bf P\}^n$ with isolated fixed points such that the collection of representations of $S^1$ on the tangent spaces at the isolated fixed points as distinct from the collection of representations occurring for any “linear” $S^1$ action on $\{\bf C\}\{\bf P\}^n$.},
author = {Petrie, Ted},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {135-150},
publisher = {Association des Annales de l'Institut Fourier},
title = {Real algebraic actions on projective spaces - A survey},
url = {http://eudml.org/doc/74119},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Petrie, Ted
TI - Real algebraic actions on projective spaces - A survey
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 2
SP - 135
EP - 150
AB - Let $G$ be a compact lie group. We introduce the set ${\bf S}_G(Y)$ for every smooth $G$ manifold $Y$. It consists of equivalence classes of pair $(X,f)$ where $f:X\rightarrow Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$. Two pairs $(X_i,f_i)$, for $i=0,1$, are equivalent if there is a $G$ homotopy equivalence $\varphi :X_0\rightarrow X_1$ such that $f_0$ is $G$ homotopic to $f_1\circ \varphi $.Properties of the set ${\bf S}_G(Y)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G=S^1$ and $Y$ is the $S^1$ manifold defined by a “linear” $S^1$ action on complex projective $n$ space ${\bf C}{\bf P}^n$, we exhibit non-trivial elements of ${\bf S}_{S^4}(Y)$ by discussing a real algebraic action of $S^1$ on ${\bf C}{\bf P}^n$ with isolated fixed points such that the collection of representations of $S^1$ on the tangent spaces at the isolated fixed points as distinct from the collection of representations occurring for any “linear” $S^1$ action on ${\bf C}{\bf P}^n$.
LA - eng
UR - http://eudml.org/doc/74119
ER -