An Introduction to Differential Manifolds by Jacques Lafontaine

By Jacques Lafontaine

This publication is an advent to differential manifolds. It provides good preliminaries for extra complex issues: Riemannian manifolds, differential topology, Lie idea. It presupposes little heritage: the reader is just anticipated to grasp easy differential calculus, and a bit point-set topology. The publication covers the most themes of differential geometry: manifolds, tangent area, vector fields, differential varieties, Lie teams, and some extra subtle issues resembling de Rham cohomology, measure conception and the Gauss-Bonnet theorem for surfaces.

Its ambition is to provide good foundations. particularly, the advent of “abstract” notions equivalent to manifolds or differential kinds is stimulated through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty workouts, a few of them effortless and classical, a few others extra subtle, may also help the newbie in addition to the extra specialist reader. ideas are supplied for many of them.

The ebook may be of curiosity to varied readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to gather a few feeling approximately this pretty theory.
The unique French textual content advent aux variétés différentielles has been a best-seller in its type in France for lots of years.

Jacques Lafontaine was once successively assistant Professor at Paris Diderot college and Professor on the college of Montpellier, the place he's shortly emeritus. His major learn pursuits are Riemannian and pseudo-Riemannian geometry, together with a few facets of mathematical relativity. along with his own learn articles, he used to be concerned with a number of textbooks and learn monographs.

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A spectral sequence is a collection of differential bigraded R-modules {Er∗,∗ , dr }, where r = 1, 2, . . ; the differentials are either all of bidegree (−r, r − 1) (for a spectral sequence of homological type) or all of bidegree (r, 1 − r) (for a spectral sequence of cohomological type) and for all p,q is isomorphic to H p,q (Er∗,∗ , dr ). p, q, r, Er+1 It is worth repeating the caveat about differentials mentioned in Chapter 1: ∗,∗ but not dr+1 . If we think of a specknowledge of Er∗,∗ and dr determines Er+1 tral sequence as a black box with input a differential bigraded module, usually E1∗,∗ , then with each turn of the handle, the machine computes a successive homology according to a sequence of differentials.

There is a spectral sequence of algebras with E2∗,∗ ∼ = V ∗ ⊗k W ∗ , as bigraded algebras, where V ∗ and W ∗ are graded algebras, and converging to H ∗ as a graded algebra. If H ∗ is an algebraic invariant of a topological space, we can think of V ∗ and W ∗ as similar invariants. ) Suppose that H ∗ is known as well as V ∗ or W ∗ . Does a spectral sequence have enough structure as an algebraic object to allow us to obtain W ∗ when V ∗ is known or vice versa? That is, can we work backward from the answer and part of the data to the rest of the data?

Since (x2 )2 ⊗y1 has total degree 5, we want z4 of degree 4 in W ∗ , with d4 (z4 ) = (x2 )2 ⊗y1 . 5. Interpreting the answer 23 takes care of (x2 )2 ⊗z4 . Further d4 (( 12 )(z4 )2 ) = (x2 )2 ⊗(y1 ⊗z4 ); this pattern continues to give the correct E∞ -term. Arguments of this sort were introduced by [Borel53]. 26). Working backward from a known answer can lead to invariants of interest. For example, in a paper on scissors congruence [Dupont82] has set up a certain spectral sequence, converging to the trivial vector space, with a known nonzero E1 -term.

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