Orthogonal transformations for which topological equivalence implies linear equivalence

W.-C. Hsiang, William Pardon
1982 Bulletin of the American Mathematical Society  
Let R t , R 2 £ 0(ri) 9 the group of orthogonal transformations of R". We say R t and R 2 are topologically (resp. linearly) equivalent if there is a homeomorphism (resp. linear automorphism) ƒ: R n -• R n such that (Of course, linear equivalence of R t with R 2 is the same as equality of the respective sets of complex eigenvalues.) The order of an orthogonal transformation is its order as an element of 0(n). The purpose of this note is to announce and discuss the proof of the following result
more » ... e following result [HP]. THEOREM A. Let R v R 2^ 0(n) have order k = /2 m , where I is odd and m > 0. Suppose that (a) JRJ and R 2 are topologically equivalent, and (b) each eigenvalue of R\ and R l 2 is either 1 or a primitive 2 m th root of unity. Then R t and R 2 are linearly equivalent.
doi:10.1090/s0273-0979-1982-15016-9 fatcat:e5cxvkqbcrgvhlhcjln45xxnya