UC Riverside

The Browder-Straus Theorem, obtained independently by S. H. Straus in the 1960s and W. Browder in the 1970's, states that for manifolds with smooth group actions, isovariant and equivariant homotopy equivalences are the same provided the spaces involved satisfy some dimensional restrictions on their fixed point sets. A proof of this theorem was presented in 2013 by R. Schultz using contributions from surgery theory. In particular, the proof requires an equivariant version of C.T.C Wall's π − π Lemma. In this dissertation we demonstrate that the dimensional restrictions of the Browder- Straus Theorem are sharp by providing examples, just outside of those dimensional restrictions, for which the conclusions of the theorem do not hold. Additionally, we discuss the implications of these examples with respect to the Equivariant π − π Lemma.