A Borsuk-Ulman type theorem for Z_4-actions - Stefanov, Simeon (Ann. Sofia Univ. Fac. Math. and Inf., 1995)
Let n=2k+1 and the sphere S^n be represented as S^n={ z=(z_1,..,z_k+1) C^k+1||z||=1 } Consider the canonical action of the group Z_4={ 1,i,-1,-i} in S^n defined by multiplication. The main result in the article is the following Borsuk-Ulam type theorem: For any continuous function f:S^n → R^1 consider the set A(f)={z S^n|f(z)=f(iz)=f(-z)=f(-iz)} Then dimA(f) geq n-3. The main corollary: For any continuous function f:S^3 → R^1 there exists z S^3 such that f(z)=f(iz)=f(-z)=f(-iz)