Homology preservation in subcomplexes Λ_i(Δ) of the barycentric subdivision

Prove that for every simplicial complex Δ and integers i ≥ −1 and −1 ≤ j ≤ i, the reduced homology groups H_j(Λ_i(Δ)) and H_j(Δ) are isomorphic, where Λ_i(Δ) is the induced subcomplex of the barycentric subdivision B(Δ) on vertices u_σ such that the link of σ in Δ has nontrivial homology in some degree ≤ i.

Background

Section 7.3 seeks the smallest subcomplex of the barycentric subdivision B(Δ) that preserves the homology of Δ up to a given degree. The subcomplex Λ_i(Δ) is defined by selecting barycentric vertices whose links in Δ have nontrivial homology in degrees up to i.

Based on computational evidence, the author conjectures that Λ_i(Δ) preserves homology up to degree i, formalized as Conjecture 7.9.