By Dominique Arlettaz

The second one Arolla convention on algebraic topology introduced jointly experts overlaying a variety of homotopy thought and $K$-theory. those lawsuits replicate either the range of talks given on the convention and the variety of promising study instructions in homotopy conception. The articles contained during this quantity contain major contributions to classical risky homotopy concept, version classification idea, equivariant homotopy conception, and the homotopy conception of fusion platforms, in addition to to $K$-theory of either neighborhood fields and $C^*$-algebras

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Riemann then artificially created barriers, until each closed curve which avoids them bounds a portion of the surface, and he cut along them. It is a specificity of dimension 2 that the surface obtained by this cutting process is automatically simply connected (it is in fact homeomorphic to a disk). Indeed, in his paper [151] published in 1904, Poincaré gave the first example of a manifold of 5 Note that here, in contrast to the situation considered in the Introduction, the loops are allowed to intersect.

Similarly, an arbitrary elliptic function, that is, a function whose derivative does not contain any other irrationality than a radical of degree two, under which the variable does not exceed the fourth degree,1 will also have the property that one may express 1 That is, the square root of a polynomial of degree at most four. © Springer International Publishing Switzerland 2016 P. 1007/978-3-319-42312-8_9 21 22 9 Abel and the New Transcendental Functions an arbitrary sum of such functions by an algebraic and a logarithmic function, provided that one establishes between the variables of those functions a certain algebraic relation.

The difference of the integrals computed on two paths with the same end-points is an integral on a closed curve. Such an integral is called a period of the form which is being integrated. 1. Replace the paths cj of its statement by other paths c0j which still satisfy the same constraints. Taken together, the paths cj followed by the paths c0j with the opposite orientation form a finite set of closed curves drawn on the Riemann surface (see the example in Fig. 1). When one changes the paths © Springer International Publishing Switzerland 2016 P.