Non-embeddability of real projective plane in R3

Patrik Zajec (2018) Non-embeddability of real projective plane in R3. EngD thesis.

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Abstract

The main goal of this thesis is to present the elementary proof for nonembeddability of real projective plane in the 3-dimensional Euclidean space and to study the embeddability of closed surfaces in general. We start with the short introduction of ideal points concept from projective geometry and present different geometrical presentations of real projective plane. Next we prove the classification theorem for closed surfaces which states that any connected closed surface is homeomorphic to the sphere, a connected sum of tori or a connected sum of real projective planes. Embeddability of orientable surfaces follows immediately from the classification of closed surfaces while the proofs for non-embeddability of non-orientable surfaces usually require techniques from algebraic topology. We prove the non-embeddability of nonorientable closed surfaces using the property of the graph K6 embedded in the Mobius strip.

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