Deformation to a Constant Loop: Brouwer's Fixed Point Theorem for D^2

We give an informal proof of the Brouwer's Fixed Point Theorem for a two-dimensional disc, using the fact that the identity loop of S^1 is not nullhomotopic (i.e., cannot be continuously deformed to a constant loop).

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A proof of this exact flavor with differential topology ingredients can be found in Guillemin/Pollack's "Differential Topology" and Milnor's "Topology from the Differentiabe Viewpoint". A proof using the fundamental group can be found in Munkres's "Topology" and Hatcher's "Algebraic Topology".