How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.

Solution 1:

Though not as easy as homological proofs, there is a known purely analytical proof that the sphere is not contractible. It follows from a purely analytical proof of Brouwer's fixed point theorem due to Milnor here.

Once Brouwer's fixed point theorem is proved, it follows that the sphere is not a retract of the closed ball and this means that the sphere can not be contractible.

• If the sphere were a retract of the ball, then the retract followed by the antipodal map on the sphere would be a fixed point free mapping of the ball into itself - contradicting Brouwer's fixed point theorem.

• A contraction may be thought of as a retraction mapping of the cone on a space into the space.

$$ S \rightarrow Cone(S) \rightarrow S$$

The closed ball is homeomorphic to the cone on the sphere.

Solution 2:

Following is taken from Fixed point theory - Dugundji, p. 95.

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