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Using algebra in geometry
Paták, Pavel ; Růžička, Pavel (advisor) ; Šmíd, Dalibor (referee) ; Blagojevic, Pavle (referee)
Using algebra in geometry Pavel Paták Department: Department of Algebra Supervisor: Mgr. Pavel Růžička, Ph.D., Department of Algebra 1 Abstract In this thesis, we develop a technique that combines algebra, algebraic topology and combinatorial arguments and provides non-embeddability results. The novelty of our approach is to examine non- embeddability arguments from a homological point of view. We illustrate its strength by proving two interesting theorems. The first one states that k-dimensional skeleton of b 2k+2 k + k + 3 -dimensional simplex does not embed into any 2k-dimensional manifold M with Betti number βk(M; Z2) ≤ b. It is the first finite upper bound for Kühnel's conjecture of non-embeddability of simplices into manifolds. The second one is a very general topological Helly type theorem for sets in Rd : There exists a function h(b, d) such that the following holds. If F is a finite family of sets in Rd such that ˜βi ( G; Z2) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1, then F has Helly number at most h(b, d). If we are only interested whether the Helly numbers are bounded or not, the theorem subsumes a broad class of Helly types theorems for sets in Rd . Keywords: Homological Non-embeddability, Helly Type Theorem, Kühnel's conjecture of non-embeddability of ske- leta of simplices into manifolds
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