equivalent statements to statement that sphere is not contractible

Let V be a normed space. Recall the definition of the sphere and the ball in V:


PropositionPlanetmathPlanetmath. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

(1) ๐•Š is not contractible;

(2) for each continous map F:๐”นโ†’๐”น there exists xโˆˆ๐”น such that Fโข(x)=x;

(3) there is no retractionMathworldPlanetmath from ๐”น onto ๐•Š.

Proof. The proof of this proposition probably can be found in some books about topologyMathworldPlanetmath. I present here the proof from my lecture due to Prof. Gโขoยดโขrniewicz.

(1)โ‡’(2) Assume there exists a continous map F:๐”นโ†’๐”น such that for each xโˆˆ๐”น we have Fโข(x)โ‰ x. Define a map H:๐•Šร—[0,1]โ†’๐•Š as follows:

H(x,t)={x-2โขtโขFโข(x)โˆฅx-2โขtโขFโข(x)โˆฅ,ifย โข0โ‰คtโ‰ค12(2-2โขt)โขx-Fโข((2-2โขt)โขx)โˆฅ(2-2โขt)โขx-Fโข((2-2โขt)โขx)โˆฅifย โข12โ‰คtโ‰ค1

Thanks to the condition Fโข(x)โ‰ x this map is well defined and it is easy to check that this is a homotopyMathworldPlanetmath from the identity map to constant map. But ๐•Š is not contractible. ContradictionMathworldPlanetmathPlanetmath.

(2)โ‡’(3) Assume there exists a retraction r:๐”นโ†’๐•Š. Define a map F:๐”นโ†’๐”น by the formulaMathworldPlanetmathPlanetmath Fโข(x)=-rโข(x). This map has no fixed pointPlanetmathPlanetmath. Contradiction.

(3)โ‡’(1) Assume that ๐•Š is contractible and take any homotopy H:๐•Šร—[0,1]โ†’๐•Š from constant map to identity map, i.e. for all xโˆˆ๐•Š we have Hโข(x,0)=x0 (for some x0โˆˆ๐•Š) and Hโข(x,1)=x. Define a map r:๐”นโ†’๐•Š as follows:

r(x)={x0,ifย โขโˆฅxโˆฅโ‰ค12Hโข(xโˆฅxโˆฅ,2โขโˆฅxโˆฅ-1)ifย โขโˆฅxโˆฅโ‰ฅ12

It is easy to see that this formula defines a retraction from ๐”น onto ๐•Š. Contradiction. โ–ก

Note that this proposition does not state that any of the conditions (1),(2),(3) hold. It only states that they are equivalent. It is well known that all of them are true if V is finite dimensional and all are false if V is infinite dimensional.

Title equivalent statements to statement that sphere is not contractible
Canonical name EquivalentStatementsToStatementThatSphereIsNotContractible
Date of creation 2013-03-22 18:07:53
Last modified on 2013-03-22 18:07:53
Owner joking (16130)
Last modified by joking (16130)
Numerical id 10
Author joking (16130)
Entry type Theorem
Classification msc 55P99