A Note on Equivariant Fixed Point Theory

be found in all classical books on the subject (for example in [4, 44]). For every x ∈ X the isotropy subgroup (also termed fixer or stabilizer) of x is Gx = {g ∈ G : gx = x}. If H ⊂ G is a subgroup of G, then the space fixed by H in X is X = {x ∈ X : Hx = x} = {x ∈ X : H ⊂ Gx}. If X and Y are G-spaces, then a G-map (i.e. an equivariant map) f : X → Y is a map which commutes with the G-action: for every g ∈ G and x ∈ X, f(gx) = gf(x). If x ∈ X and Gx is its isotropy, g ∈ Gx =⇒ gfx = fgx = fx =⇒ g ∈ Gfx, thus Gfx ⊃ Gx. This implies that if f is equivariant, then for every H ⊂ G, fX ⊂ Y H . The restriction of f to the fixed subspace X is denoted by f : X → Y H . A G-homotopy H : f0 ∼ f1 is a G-map H : X × I → Y , where the action of G on X× I is trivial on the I-component. In equivariant