The Dold$-$Thom theorem via factorization homology
The Dold$-$Thom theorem is a classical result in algebraic topology relating homotopy and homology. It states that for a nice topological space $M$, there are isomorphisms between the homotopy groups of the infinite symmetric product of $M$ and the reduced homology groups of $M$ itself: $\pi_*({\mathrm{Sym}}(M; A)) \cong \widetilde{H}_*(M;A).$ The classical proof appeals to the Eilenberg$-$Steenrod axioms to verify the composition of functors $\pi_*({\mathrm{Sym}}(−; A))$ determines the same homology theory as reduced singular homology. In this talk, we will outline a new, direct proof of the Dold$-$Thom theorem which avoids the Eilenberg$-$Steenrod axioms entirely. The heart of this proof is a local-to-global argument allowing us to recognize the infinite symmetric product as an instance of factorization homology.