By Martin Wirsing, Jan A. Bergstra
The right kind therapy and selection of the elemental information constructions is a crucial and complicated half within the means of application development. Algebraic tools supply concepts for info abstraction and the based specification, validation and research of information constructions. This quantity originates from a workshop equipped inside ESPRIT venture 432 METEOR, An built-in Formal method of business software program improvement, held in Mierlo, The Netherlands, September 1989. the quantity contains 5 invited contributions in accordance with workshop talks given through A. Finkelstein, P. Klint, C.A. Middelburg, E.-R. Olderog, and H.A. Partsch. Ten extra papers via individuals of the METEOR staff are in keeping with talks given on the workshop. The workshop used to be a successor to an prior one held in Passau, Germany, June 1987, the lawsuits of which have been released as Lecture Notes in machine technological know-how, Vol. 394.
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Let us look at two of the simplest examples: the closed disk D and the closed annulus A. On D, any orientation-reversing homeomorphism f induces a degree −1 map on S 1 = ∂D, and from this follows that f is not isotopic to the identity. However, the straight-line homotopy gives a homotopy between f and the identity. On A = S 1 × I, the orientation-reversing map that fixes the S 1 factor and reflects the I factor is homotopic but not isotopic to the identity. It turns out that these two examples are the only examples of homotopic homeomorphisms that are not isotopic.
Proof. Assume χ(S) ≤ 0, so the universal cover S is homeomorphic to R2 (the case of χ(S) > 0 is an exercise). Let p : S → S be the covering map. Suppose the lifts α and β of α and β intersect in at least two points. It follows that there is an embedded disk D0 in S bounded by one subarc of α and one subarc of β. By compactness and transversality, the intersection (p−1 (α)∪ p−1 (β))∩ D0 is a finite graph, if we think of the intersection points as vertices. 3). Denote the two “vertices” of D by v1 and v2 , and the two “edges” of D by α1 and β1 .
We know that π1 (T 2 ) ≈ Z2 , and, if we base π1 (T 2 ) at the image of the origin, one way to get a representative for (p, q) as a loop in T 2 is to take the straight line from (0, 0) to (p, q) in R2 and project it to T 2 . Let γ be any oriented simple closed curve in T 2 . Up to homotopy, we can assume that γ passes through the image in T 2 of (0, 0) in R2 . Any path lifting of γ to R2 based at the origin terminates at some integral point (p, q). There is then a homotopy from γ to the standard straight-line representative of (p, q) ∈ π1 (T 2 ); indeed, the straight-line homotopy from the lift of γ to the straight line through (0, 0) and (p, q) is equivariant with respect to the group of deck transformations and thus descends to the desired homotopy.
Algebraic Methods: Theory, Tools and Applications by Martin Wirsing, Jan A. Bergstra