By M. Bourlakis. S. Papagiannidis. Helen Fox
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Additional info for E-Consumer Behaviour: Past, Present and Future Trajectories of an Evolving Retail Revolution
The starting point is the case of nonseparating simple closed curves, and the inductive step is Example 5: cutting along the first few arcs, the next arc becomes a nonseparating arc on the cut surface. Note that Example 1 is the case k = 2. One can also prove by induction that every chain in Sg of even length is nonseparating, and so such chains must be topologically equivalent. We remark that the homeomorphism representing the change of coordinates in each of the six examples above can be taken to be orientation preserving.
The way we will compute Mod(S0,3 ) is to understand its action on some fixed arc in S0,3 . The surface obtained by cutting S0,3 along this arc is a punctured disk, and so we will be able to apply the Alexander lemma. This is in general how we use the cutting procedure for surfaces in order to perform inductive arguments. In this section it will be convenient to think of S0,3 as a sphere with three marked points (instead of three punctures). In order to determine Mod(S0,3 ) we first need to understand simple proper arcs in S0,3 .
44 CHAPTER 1 In fact a stronger, relative result holds: if two homeomorphisms are homotopic relative to ∂S then they are isotopic relative to ∂S. 8. 12 also holds when S has finitely many marked points. In that case, we need to expand our list of counterexamples to include a sphere with one or two marked points. 2 H OMEOMORPHISMS VERSUS DIFFEOMORPHISMS It is sometimes convenient to work with homeomorphisms and sometimes convenient to work with diffeomorphisms. For example it is easier to construct the former but we can apply differential topology to the latter.