We present a composite framework to explore deviations from ΛCDM by independently parameterizing the background expansion and the growth of cosmic structures. Using current observational data, we find strong indications of deviation from ΛCDM in the background expansion, consistent with recent DESI DR2 results, while no significant departure is observed in the growth sector. This is primarily due to limitations in current growth measurements, making it difficult to distinguish between evolving dark energy and modified gravity. We further show how future high-precision surveys like Euclid can break this degeneracy and decisively test such scenarios.

In this talk, we will study the theory of the Euler class group, developed to study projective modules over Noetherian rings and to address the fundamental question of whether a projective module $P$ splits off a free summand of rank one. This question is motivated by obstruction theory in topology, which studies when vector bundles on smooth real manifolds admit nowhere vanishing section. We will define the Euler class group $E(A)$ for a ring $A$, and explain how to associate to a given projective module over $A$ a class in $E(A)$ whose vanishing provides an affirmative answer to our splitting question. As an application of Euler class theory, we will prove an obstruction criteria in terms of generic sections for projective modules over real affine varieties. This talk is primarily based on the foundational paper in this area by Bhatwadekar and Sridharan titled ``The Euler class group of a Noetherian ring", Compositio Math. 122 (2000).