A Simple Proof of a Generalized No Retraction Theorem

نویسنده

Ethan D. Bloch

چکیده

The world does not need yet another proof of the classical no retraction theorem (NRT) and its equivalent partner the Brouwer fixed point theorem (BFPT)— many lovely elementary proofs are widely known. What does merit a new proof, however, is a much less well-known generalization of the NRT to a broader class of topological spaces than only those that are homeomorphic to balls (which is what the NRT and BFPT are traditionally about). The purpose of this paper is to state and prove this generalized NRT in the 2-dimensional case. In particular, we will show that a version of the NRT, when appropriately stated, can be proved for the class of all topological spaces that are homeomorphic to the underlying spaces of finite 2-dimensional simplicial complexes. (As the reader can verify by finding examples, the BFPT does not generalize to all such spaces, and hence the equivalence of the NRT and BFPT also does not generalize to all such spaces.) If one were interested only in the classical NRT, our proof could be simplified even further to give a particularly low-tech proof of that theorem, though we omit the details. To remind the reader of what we are generalizing, we state the classical 2-dimensional versions of the NRT and the BFPT. The exact analogs of both theorems hold in higher dimensions, though we will not discuss them in this note. We use the notations D2 and S1 to denote the unit ball and the unit circle respectively in R2; that is,

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