From Smith Theory to the Dold–Thom Theorem: A Unified Study of Symmetry in Algebraic Topology

Symmetry in topological spaces, expressed through continuous group actions, underlies some of the deepest invariants in algebraic topology. This work presents a unified framework connecting fixed-point theory, prime-order group actions, equivariant cohomology, symmetric products, and covering symmetries, areas that have traditionally been developed independently. All results are developed within the category of compactly generated Hausdorff G-spaces and equivariant maps, where G is a finite topological group acting continuously on a topological space X (with G taken as a compact torus (S¹)ⁿ in the localisation result). The principal tools are the Borel construction EG ×_G X, Smith exact triangles over ℤ/pℤ-module complexes, the infinite symmetric product functor, and the monodromy action of the fundamental group on the universal cover. We present several fundamental results linking fixed-point theory, Smith theory, equivariant cohomology, symmetric products, and covering spaces. In fixed-point theory, the Lefschetz Fixed-Point Theorem is proved and refined equivariantly, showing that a non-zero Lefschetz number forces a G-invariant fixed-point set. In Smith theory, the Smith Inequality bounds the mod-p homological dimension of X^G relative to X, and the Smith–Floyd Theorem yields sharp Lusternik–Schnirelmann category estimates for free ℤ₂-actions. For equivariant cohomology, an Equivariant Künneth Formula and a Localisation Theorem show that torus-equivariant cohomology localises to fixed-point data after inverting the augmentation ideal. The Dold–Thom Theorem is proved in full, identifying π_n(SP^∞(X)) with H_n(X; ℤ) via the Eilenberg–Steenrod axioms. Finally, the Galois Correspondence between subgroups of π₁(B) and connected covering spaces is established, with the Deck Group Isomorphism as its centerpiece. These results demonstrate that equivariant symmetry acts as a unifying principle across algebraic topology. Natural directions for further work include Smith theory for non-abelian actions, motivic equivariant cohomology, and symmetric products of non-simply connected spaces.