Lefschetz fixed-point theorem - Revision history

Vipul: /* Corollaries */

2011-07-27T17:18:52Z

Corollaries

← Older revision		Revision as of 17:18, 27 July 2011
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	* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]		* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]
	* [[~~Acyclic~~ compact polyhedron has fixed-point property]]~~: More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property~~		* [[Rationally acyclic compact polyhedron has fixed-point property]]
	* [[Euler characteristic of compact connected nontrivial Lie group is zero]]: [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic\|zero]]		* [[Euler characteristic of compact connected nontrivial Lie group is zero]]: [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic\|zero]]
	* [[Self-map of sphere of degree greater than one has a fixed point]]: Any map from a sphere to itself of degree greater than 1 must have a fixed point		* [[Self-map of sphere of degree greater than one has a fixed point]]: Any map from a sphere to itself of degree greater than 1 must have a fixed point

Vipul: /* Corollaries */

2011-07-27T17:17:37Z

Corollaries

← Older revision		Revision as of 17:17, 27 July 2011
Line 6:		Line 6:

	* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]		* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]
	* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property		* [[Acyclic compact polyhedron has fixed-point property]]: More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property
	* ~~The~~ [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic\|zero]]		* [[Euler characteristic of compact connected nontrivial Lie group is zero]]: [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic\|zero]]
	* Any map from a sphere to itself of degree greater than 1 must have a fixed point		* [[Self-map of sphere of degree greater than one has a fixed point]]: Any map from a sphere to itself of degree greater than 1 must have a fixed point

Vipul: 4 revisions

2008-05-11T19:48:20Z

4 revisions

← Older revision	Revision as of 19:48, 11 May 2008
(No difference)

Vipul: /* Corollaries */

2007-10-27T22:42:14Z

Corollaries

← Older revision		Revision as of 22:42, 27 October 2007
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	* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]		* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]
	* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property		* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property
	* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic\|zero]]		* The [[Euler characteristic]] of any nontrivial compact connected Lie group is [[space with zero Euler characteristic\|zero]]
			* Any map from a sphere to itself of degree greater than 1 must have a fixed point

Vipul: /* Corollaries */

2007-10-27T22:40:54Z

Corollaries

← Older revision		Revision as of 22:40, 27 October 2007
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	==Corollaries==		==Corollaries==

	* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer's fixed-point theorem]]		* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer fixed-point theorem]]
	* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property		* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property
	* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic\|zero]]		* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic\|zero]]

Vipul: /* Corollaries */

2007-10-27T22:40:35Z

Corollaries

← Older revision		Revision as of 22:40, 27 October 2007
Line 5:		Line 5:
	==Corollaries==		==Corollaries==

	* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property		* Any [[contractible space\|contractible]] compact polyhedron has the [[fixed-point property]]. In particular, every disc has the fixed-point property, which is [[Brouwer's fixed-point theorem]]
			* More generally, every [[acyclic space\|acyclic]] compact polyhedron has the fixed-point property
	* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic\|zero]]		* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic\|zero]]

Vipul at 22:39, 27 October 2007

2007-10-27T22:39:52Z

New page

==Statement==

If the [[Lefschetz number]] of a map from a [[compact polyhedron]] (viz a [[compact space]] that is also a [[polyhedron]]) to itself is nonzero, then the map has a fixed point.

==Corollaries==

* Any [[contractible space|contractible]] compact polyhedron has the [[fixed-point property]]. More generally, every [[acyclic space|acyclic]] compact polyhedron has the fixed-point property
* The [[Euler characteristic]] of any compact connected Lie group is [[space with zero Euler characteristic|zero]]