Spectral Homotopy and the Spectral Fundamental Group

Abstract

In this paper, we introduce an algebraic-topological invariant for commutative pm-rings, termed the spectral fundamental group, which is denoted by $\pi_{k}^{alg}(A)$. This group is defined via homotopy classes of loops within the space of induced spectral maps, which are generated by the $k$-algebra endomorphism monoid of the ring. We establish foundational properties of this invariant, proving that $\pi_{k}^{alg}(A)$ is an abelian group that naturally respects direct products and admits natural morphisms with respect to fully invariant subrings. Further, we establish an explicit isomorphism between the spectral fundamental group of certain continuous function rings and the classical fundamental group of their associated topological mapping spaces. Finally, utilizing a generalized dual number construction, we present an explicit example of a pm-ring that cannot be embedded into any function ring over a field of characteristic zero, yet possesses a nontrivial spectral fundamental group. This demonstrates that $\pi_{k}^{alg}(A)$ captures homotopical dynamics that are intrinsically algebraic.