Fixed-point property - Revision history

Vipul: /* Manifolds without boundary */

2011-07-28T02:23:52Z

Manifolds without boundary

← Older revision		Revision as of 02:23, 28 July 2011
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	\| [[quaternionic projective space]] <math>\mathbb{P}^n(\mathbb{H}), n \ge 1</math> \|\| <math>4n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[quaternionic projective space has fixed-point property iff it has even quaternionic dimension]].		\| [[quaternionic projective space]] <math>\mathbb{P}^n(\mathbb{H}), n \ge 1</math> \|\| <math>4n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[quaternionic projective space has fixed-point property iff it has even quaternionic dimension]].
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| ~~<math>g = 0,1,g \ge 3</math>:~~ No~~<br><math>g = 2</math>: Not sure~~ \|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| No \|\|
	\|-		\|-
	\| [[torus]] <math>T^n, n \ge 1</math>, product of <math>n</math> [[circle]]s \|\| <math>n</math> \|\| No \|\| Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.		\| [[torus]] <math>T^n, n \ge 1</math>, product of <math>n</math> [[circle]]s \|\| <math>n</math> \|\| No \|\| Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.

Vipul: /* Manifolds without boundary */

2011-07-28T02:03:45Z

Manifolds without boundary

← Older revision		Revision as of 02:03, 28 July 2011
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	\| [[quaternionic projective space]] <math>\mathbb{P}^n(\mathbb{H}), n \ge 1</math> \|\| <math>4n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[quaternionic projective space has fixed-point property iff it has even quaternionic dimension]].		\| [[quaternionic projective space]] <math>\mathbb{P}^n(\mathbb{H}), n \ge 1</math> \|\| <math>4n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[quaternionic projective space has fixed-point property iff it has even quaternionic dimension]].
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g ~~\ge~~ 2</math>: ~~Yes (''need to verify'')~~\|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1,g \ge 3</math>: No<br><math>g = 2</math>: Not sure \|\|
	\|-		\|-
	\| [[torus]] <math>T^n, n \ge 1</math>, product of <math>n</math> [[circle]]s \|\| <math>n</math> \|\| No \|\| Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.		\| [[torus]] <math>T^n, n \ge 1</math>, product of <math>n</math> [[circle]]s \|\| <math>n</math> \|\| No \|\| Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.

Vipul: /* Manifolds without boundary */

2011-07-28T02:02:46Z

Manifolds without boundary

← Older revision		Revision as of 02:02, 28 July 2011
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	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has even complex dimension]].		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has even complex dimension]].
			\|-
			\| [[quaternionic projective space]] <math>\mathbb{P}^n(\mathbb{H}), n \ge 1</math> \|\| <math>4n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[quaternionic projective space has fixed-point property iff it has even quaternionic dimension]].
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|

Vipul: /* Manifolds without boundary */

2011-07-27T19:13:28Z

Manifolds without boundary

← Older revision		Revision as of 19:13, 27 July 2011
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	\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).		\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).
	\|-		\|-
	\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has ~~odd~~ dimension]]		\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has even dimension]]
	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has ~~odd~~ complex dimension]].		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has even complex dimension]].
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|

Vipul: /* Manifolds without boundary */

2011-07-27T19:11:58Z

Manifolds without boundary

← Older revision		Revision as of 19:11, 27 July 2011
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	\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has odd dimension]]		\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has odd dimension]]
	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has odd complex dimension]]. Explicitly, for <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has odd complex dimension]].
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|

Vipul: /* Manifolds without boundary */

2011-07-27T18:20:07Z

Manifolds without boundary

← Older revision		Revision as of 18:20, 27 July 2011
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	\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).		\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).
	\|-		\|-
	\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has odd dimension]]. For <math>n</math> even, follows from [[rationally acyclic compact polyhedron has fixed-point property]]. For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>a_1^2 + a_2^2 = a_3^2 + a_4^2 = \dots = a_n^2 + a_{n+1}^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has odd dimension]]
	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has odd complex dimension]]. Explicitly, for <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has odd complex dimension]]. Explicitly, for <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.

Vipul: /* Manifolds without boundary */

2011-07-27T18:03:29Z

Manifolds without boundary

← Older revision		Revision as of 18:03, 27 July 2011
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	\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).		\| [[sphere]] <math>S^n,n \ge 1</math> \|\| <math>n</math> \|\| No \|\| The antipodal map is a fixed-point-free self-map (in fact, it's a self-homeomorphism without fixed points).
	\|-		\|-
	\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> even, follows from [[rationally acyclic compact polyhedron has fixed-point property]]. For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>a_1^2 + a_2^2 = a_3^2 + a_4^2 = \dots = a_n^2 + a_{n+1}^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[real projective space has fixed-point property iff it has odd dimension]]. For <math>n</math> even, follows from [[rationally acyclic compact polyhedron has fixed-point property]]. For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>a_1^2 + a_2^2 = a_3^2 + a_4^2 = \dots = a_n^2 + a_{n+1}^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.
	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| ~~For~~ <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| See [[complex projective space has fixed-point property iff it has odd complex dimension]]. Explicitly, for <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|

Vipul: /* Facts */

2011-07-27T18:00:26Z

Facts

← Older revision		Revision as of 18:00, 27 July 2011
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	==Facts==		==Facts==

			===Fixed-point theorems===

	* [[Lefschetz fixed-point theorem]]		* [[Lefschetz fixed-point theorem]]
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	In general, we combine the Lefschetz fixed-point theorem with the structure of the [[cohomology ring]] of the space to determine whether or not it has the fixed-point property. For instance, we can show that [[complex projective space]] in even dimensions has the fixed-point property, by combining the Lefschetz fixed-point theorem with the fact that the trace on the <math>(2k)^{th}</math> homology is <math>d^k</math> where <math>d</math> is the trace on the second homology.		In general, we combine the Lefschetz fixed-point theorem with the structure of the [[cohomology ring]] of the space to determine whether or not it has the fixed-point property. For instance, we can show that [[complex projective space]] in even dimensions has the fixed-point property, by combining the Lefschetz fixed-point theorem with the fact that the trace on the <math>(2k)^{th}</math> homology is <math>d^k</math> where <math>d</math> is the trace on the second homology.

			===Products===

			The product of any topological space that does ''not'' satisfy the fixed-point property with any nonempty topological space gives a space that does ''not'' satisfy the fixed-point property.

	==Metaproperties==		==Metaproperties==

Vipul: /* Manifolds without boundary */

2011-07-27T17:59:23Z

Manifolds without boundary

← Older revision		Revision as of 17:59, 27 July 2011
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	\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> even, follows from [[rationally acyclic compact polyhedron has fixed-point property]]. For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>a_1^2 + a_2^2 = a_3^2 + a_4^2 = \dots = a_n^2 + a_{n+1}^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[real projective space]] <math>\mathbb{P}^n(\R), n \ge 1</math> \|\| <math>n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> even, follows from [[rationally acyclic compact polyhedron has fixed-point property]]. For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-a_2:a_1:-a_4:a_3:\dots:-a_{n+1}:a_n]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>a_1^2 + a_2^2 = a_3^2 + a_4^2 = \dots = a_n^2 + a_{n+1}^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.
	\|-		\|-
	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because ~~solviing~~ the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solving the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes \|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes (''need to verify'')\|\|
			\|-
			\| [[torus]] <math>T^n, n \ge 1</math>, product of <math>n</math> [[circle]]s \|\| <math>n</math> \|\| No \|\| Follows from the fact that it's a nontrivial Lie group, so multiplication by a non-identity element. Also, we can use the coordinate-wise antipodal map.
			\|-
			\| [[product of spheres]] <math>S^{m_1} \times S^{m_2} \times \dots S^{m_r}</math> \|\| <math>m_1 + m_2 + \dots + m_r</math> \|\| No \|\| Take the antipodal map for each coordinate sphere.
	\|}		\|}

Vipul: /* Manifolds without boundary */

2011-07-27T17:56:10Z

Manifolds without boundary

← Older revision		Revision as of 17:56, 27 July 2011
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	\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solviing the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.		\| [[complex projective space]] <math>\mathbb{P}^n(\mathbb{C}), n \ge 1</math> \|\| <math>2n</math> \|\| Even <math>n</math>: Yes<br>Odd <math>n</math>: No \|\| For <math>n</math> odd, we use the mapping <math>[a_1:a_2:a_3:a_4:\dots:a_n:a_{n+1}] \mapsto [-\overline{a_2}:\overline{a_1}:-\overline{a_4}:\overline{a_3}:\dots:-\overline{a_{n+1}}:\overline{a_n}]</math>. Note that the map has no fixed points because solviing the fixed point condition gives <math>\|a_1\|^2 + \|a_2\|^2 = \|a_3\|^2 + \|a_4\|^2 = \dots = \|a_n\|^2 + \|a_{n+1}\|^2 = 0</math>, forcing <math>a_1 = a_2 = \dots = a_n = a_{n+1} = 0</math>.
	\|-		\|-
	\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br>g \ge 2</math>: Yes \|\|		\| [[compact orientable surface]] of genus <math>g \ge 0</math> \|\| 2 \|\| <math>g = 0,1</math>: No<br><math>g \ge 2</math>: Yes \|\|
	\|}		\|}