By Jean Dieudonné

A vintage on hand back! This booklet lines the historical past of algebraic topology starting with its production through Henry Poincaré in 1900, and describing intimately the $64000 principles brought within the thought earlier than 1960. In its first thirty years the sphere appeared restricted to functions in algebraic geometry, yet this replaced dramatically within the Nineteen Thirties with the construction of differential topology through Georges De Rham and Elie Cartan and of homotopy thought by way of Witold Hurewicz and Heinz Hopf. The impression of topology started to unfold to progressively more branches because it progressively took on a relevant position in arithmetic. Written via a world-renowned mathematician, this booklet will make interesting studying for a person operating with topology.

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**Sample text**

One of the most striking facts about the Poincar´e metric on the disk is that it turns the disk into a complete metric space. How could this be? The boundary is missing! The reason that the disk is complete in the Poincar´e metric is the same as the reason that the plane is complete in the Euclidean metric: the boundary is inﬁnitely far away. We now prove this assertion. 16. The unit disk D, when equipped with the Poincar´e metric, is a complete metric space. 3 A Geometric View of the Schwarz Lemma 43 Proof.

One of the most striking facts about the Poincar´e metric on the disk is that it turns the disk into a complete metric space. How could this be? The boundary is missing! The reason that the disk is complete in the Poincar´e metric is the same as the reason that the plane is complete in the Euclidean metric: the boundary is inﬁnitely far away. We now prove this assertion. 16. The unit disk D, when equipped with the Poincar´e metric, is a complete metric space. 3 A Geometric View of the Schwarz Lemma 43 Proof.

Let ρ be the Poincar´e metric on the disk D. Let h : D → D be a conformal self-map of the disk. Then h is an isometry of the pair (D, ρ) with the pair (D, ρ). Proof. We have that h∗ ρ(z) = ρ(h(z)) · h (z) . We now have two cases: (i) If h is a rotation, then h(z) = μ · z for some unimodular constant μ ∈ C. So h (z) = 1 and 1 1 − μz h∗ ρ(z) = ρ(h(z)) = ρ(μz) = 2 = 1 1− z 2 as desired. (ii) If h is a M¨ obius transformation, then h(z) = z−a , 1 − az some constant a ∈ D. But then h (z) = 1− a 2 1 − az 2 and h∗ ρ(z) = ρ = z−a 1 − az 1 1− z−a 1−az · h (z) 2 · 1− a 2 1 − az 2 1− a 2 1 − az 2 − z − a 2 1− a 2 = 2 1− z − a 2+ a 1 = 1− z 2 = ρ(z).

### A History Of Algebraic And Differential Topology, 1900-1960 by Jean Dieudonné

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