By Dominique Arlettaz

ISBN-10: 082183696X

ISBN-13: 9780821836965

ISBN-10: 3019815835

ISBN-13: 9783019815834

ISBN-10: 7119964534

ISBN-13: 9787119964539

ISBN-10: 8619866036

ISBN-13: 9788619866033

The second one Arolla convention on algebraic topology introduced jointly experts protecting quite a lot of homotopy conception and $K$-theory. those complaints mirror either the range of talks given on the convention and the range of promising learn instructions in homotopy thought. The articles contained during this quantity contain major contributions to classical volatile homotopy conception, version classification thought, equivariant homotopy concept, and the homotopy thought of fusion structures, in addition to to $K$-theory of either neighborhood fields and $C^*$-algebras

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**Additional resources for An Alpine Anthology of Homotopy Theory**

**Example text**

2). Z/ such that (i) …A 1 …t D 0 t (ii) i …A 1 … > 0. ii/ are called Riemann relations. X //, the C-vector space of holomorphic 1-forms on a compact Riemann surface X of genus g, has dimension g. X; Z/ is a free abelian group of rank 2g. X //, namely W ! 7! which does by Stokes’ theorem not depend on the choice of the representative. This map is injective. X; Z/ Jacobians of Riemann surfaces of genus g 1 can be canonically polarized and this polarization is principal (see [BL04], p. 317, for details).

Nevertheless, the group of automorphisms of XD might be much bigger than the trivial group. An explicit example of an automorphism of X8 not stemming from an automorphism of H H is described in [vdG88], Chap. 2. For a given D, it is in general an unsolved problem how to explicitly write down all automorphisms of XD . Hilbert Modular Forms. Hilbert modular forms are a higher dimensional analogue of elliptic modular forms. Many parts of the theory of elliptic modular forms can be translated into the language of Hilbert modular forms.

Let us explain how this gives rise to a metric with the properties mentioned previously. / denote the zeroes of !. / a flat (Riemannian) metric is given by pulling back the Euclidean metric of C via the charts. / leads to a singularity of this metric. The total angle around a singularity, called the cone angle, is an integer multiple of 2 . If the form ! g. [Zor06], Sect. 3). A geodesic segment connecting two singular points is called saddle connections. Equivalently, flat surfaces arise from gluing rational angled planar polygons by parallel translations along their faces.

### An Alpine Anthology of Homotopy Theory by Dominique Arlettaz

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