By Jean Dieudonné
A vintage on hand back! This e-book strains the background of algebraic topology starting with its construction by means of Henry Poincaré in 1900, and describing intimately the real principles brought within the concept ahead of 1960. In its first thirty years the sphere appeared restricted to purposes in algebraic geometry, yet this replaced dramatically within the Nineteen Thirties with the construction of differential topology by way of Georges De Rham and Elie Cartan and of homotopy idea through Witold Hurewicz and Heinz Hopf. The impression of topology started to unfold to progressively more branches because it steadily took on a vital position in arithmetic. Written by way of a world-renowned mathematician, this e-book will make intriguing examining for someone operating with topology.
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Additional info for A History Of Algebraic And Differential Topology, 1900-1960
Y = −(f + s)y + r, y(0) = 0. Consider any nontrivial solution x of the homogeneous equation x =−(1/2)(f +s)x. If u is the function satisfying u = 2r/x2 , u(0) = 0, then y = (1/2)ux2 . But u ≥ 0, so that y ≥ 0, and s ≤ f . This contradicts the fact that s is deﬁned for all time, since f (t) → −∞ as t −1/s(0). The condition that the Jacobi ﬁelds in V span the normal space of c at every point is essential: Consider for example R2 with the standard metric, and a geodesic c in R2 . If E is a parallel ﬁeld orthogonal to c, then t → J(t) := tE(t) deﬁnes a non-parallel Jacobi ﬁeld, and represents an (n − 1)-dimensional space of Jacobi ﬁelds satisfying all the conditions of the theorem except for the one above.
Submersions, Foliations, and Metrics (vii) Denote by H : T M −→ H ⊂ T M the orthogonal projection onto the horizontal subbundle. It is interesting to note that the A and S tensors are just the covariant derivatives in horizontal and vertical directions of this projection restricted to H, cf. . More precisely, for horizontal x and vertical u, ∇x H|H = Ax , Sx u = −(∇u H)x. In fact, for a horizontal ﬁeld Y that evaluates to the vector y at the foot-point of x , ∇x (HY ) = (∇x H)Y − H∇x Y , so that (∇x H)Y = ∇x (HY ) − H∇x Y = Ax y, and a similar formula holds for ∇u H.
1, K(r) = 3 [X, Y ], T 4 2 = 3 . (1 + r2 )2 32 Chapter 1. 12) and summing over an orthonormal basis of the vertical space implies that for basic X, Y , 2 div AX Y = − tr d∇ SX,Y . 1, which asserts that the mean curvature κ of the foliation satisﬁes −dκ(X, Y ) = 2 div AX Y : since κ is just the trace of S, the above identity then follows from tr ◦ d∇ = d◦ tr, which itself reﬂects the fact that the trace operator is a parallel section of the bundle (End T M )∗ , see for example . (vi) The reader may have noticed that some of the curvature formulae in this section seem to diﬀer substantially from those in  or , even allowing for the diﬀerence in notation.