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Explore advanced concepts in combinatorial topology through this 51-minute conference talk that examines face-number-related invariants of simplicial complexes, with particular focus on normal 3-pseudomanifolds containing limited singularities. Delve into the mathematical invariant g₂, defined for normal d-pseudomanifolds as g₂(K):=f₁(K)-(d+1)f₀(K)+C(d+2,2), where f₀ and f₁ represent vertex and edge counts respectively. Learn about D.W. Walkup's classical result proving that for closed, connected triangulated 3-manifolds, the inequality f₁(Δ)-4f₀(Δ)+10 ≥ 0 holds, establishing g₂(Δ) ≥ 0. Understand how singular vertices are characterized in normal d-pseudomanifolds when |lk(v,K)| ≢ S^(d-1), where lk(v,K) denotes the vertex link. Discover two powerful combinatorial tools—vertex folding and edge folding—and their applications in characterizing normal pseudomanifolds across dimensions 3 and higher. Examine specific results concerning normal 3-pseudomanifolds containing at most two singularities under prescribed upper bounds on g₂, gaining insight into the rich combinatorial structures that enable these sophisticated mathematical operations and classifications.
