A topologist's view of the Dunford-Schwartz proof of the Brouwer fixed-point theorem. (Q5949261)

Here the author notes that the Dunford-Schwartz proof of Brouwer's fixed point theorem [\textit{N. Dunford} and \textit{J. T. Schwartz}, Linear operators. Part I, Reprint, Wiley, New York (1988; Zbl 0635.47001) (Chapter 5, Section 12)] is essentially the standard topological proof used in de Rham cohomology theory instead of simplicial homology theory. \textit{G.-C. Rota} [Indiscrete thoughts, Birkhäuser Boston (1997; Zbl 0862.00005)] wrote that this proof was submitted for publication in a journal, but was rejected because it did not use homology theory, but used some determinantal identity instead. But at the time it was written differential forms were not common knowledge and notations and terminologies of differential forms were not used. The author translates the Dunford-Schwartz proof into the language of the calculus of differential forms. For example, the determinantal identity is shown to be a direct consequence of the closedness of the volume form of \(\mathbb R^n\). By this translation, the author's statement is clearly explained.