By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski
As a part of the medical job in reference to the seventieth birthday of the Adam Mickiewicz collage in Poznan, a global convention on algebraic topology was once held. within the ensuing lawsuits quantity, the emphasis is on gigantic survey papers, a few provided on the convention, a few written therefore.
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Extra info for Algebraic Topology, Poznan 1989
M3 is a 3-manifold if its Euler characteristic is 0, [ST801. The Euler characteristic can be obtained from the quotient 3-ball complex Q, obtained from B3 by the identifications in the cellular embedding (G, S2). Let v', f', and e' be the number of 0-cells, 2-cells and 1-cells of Q, respectively. Note that Q has only one 3-cell. Thus, to say that M3 has Euler characteristic 0 means the same as v' + f' = e' + 1. We need to show that this equality is equivalent to the one which characterizes 3-gems for H*: vH.
Note that ET VT = 4vG, since each vertex appears in 4 distinct triballs. Also, ET br = 2bG, because each bigon appears in two distinct triballs . Thus, 2tG = ET(bT - vT/2) _ 2bG - 2vG , or VG + tG = bG. Conversely, suppose that G is not a 3 -gem. Then some 3-residue T' induces a surface ST, which is not a 2-sphere. Therefore,X(Sr') = bT, - vT,/2 is less than 2. Since for each 3-residue T, br - vT/2 < 2, it follows that ET(br - vT/2) < 2tG. As the sum on the left is 2bG - 2vG, we get bG < VG + tG.
Here are these constructions for the surfaces associated to the (2 + 1)-graphs of Fig. 1. Surfaces from (2 + 1) -Graphs 17 52 e f g h RP2 Fig. 18: The surfaces induced by the 2-gems of Fig 1 From (2+1)-graph, we can get at once the Euler characteristic of its surface and whether the surface is orientable or not. This means that we can topologically decide which is the surface (without constructing it explicitly). Proposition 1 The Euler characteristic of the surface S associated with a (2 + 1)graph G with v vertices and b bigons is X = b - v/2.
Algebraic Topology, Poznan 1989 by Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski