# Real algebraic actions on projective spaces - A survey

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 2, page 135-150
- ISSN: 0373-0956

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topPetrie, 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 -

## References

top- [1] M. F. ATIYAH and R. BOTT, A Lefschitz fixed point formula for elliptic complexes II, Applications, Vol. 88 (1968), 451-491. Zbl0167.21703MR38 #731
- [2] M. F. ATIYAH and G. SEGAL, Equivarient K-theory and completion, J. Diff. Geom. (3) (1968), 1-18. Zbl0215.24403MR41 #4575
- [3] M. F. ATIYAH and I. SINGER, The index of elliptic operators I, III. Annals of Math. (2) 87 (1968), 484-530 and 546-604. Zbl0164.24001MR38 #5243
- [4] T. PETRIE, S1 actions on homotopy complex projective spaces. Bull. AMS (78) 2, March 1972, 105-153. Zbl0247.57010MR45 #6029
- [5] T. PETRIE, Exotic S1 actions on CP3 and related to appear. Zbl0243.57020
- [6] T. PETRIE, Torus actions on homotopy complex projective spaces to appear. Zbl0262.57021
- [7] T. PETRIE, Real algebraic S' actions on complex projective spaces to appear. Zbl0264.57015

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