By David Spring

This booklet presents a accomplished research of convex integration idea in immersion-theoretic topology. Convex integration thought, built initially through M. Gromov, presents common topological tools for fixing the *h*-principle for a wide selection of difficulties in differential geometry and topology, with purposes additionally to PDE conception and to optimum keep watch over thought. even though topological in nature, the idea relies on an actual analytical approximation outcome for greater order derivatives of features, proved via M. Gromov. This e-book is the 1st to offer an exacting checklist and exposition of all the easy ideas and technical result of convex integration idea in better order jet areas, together with the speculation of iterated convex hull extensions and the speculation of relative *h*-principles. A moment characteristic of the booklet is its special presentation of purposes of the overall idea to themes in symplectic topology, divergence unfastened vector fields on 3-manifolds, isometric immersions, absolutely genuine embeddings, underdetermined non-linear platforms of PDEs, the relief theorem in optimum keep an eye on idea, in addition to purposes to the conventional immersion-theoretical issues similar to immersions, submersions, *k*-mersions and loose maps.

The e-book should still turn out worthy to graduate scholars and to researchers in topology, PDE concept and optimum keep watch over idea who desire to comprehend the *h*-principle and the way it may be utilized to resolve difficulties of their respective disciplines.

------ reports

The first 8 chapters of Spring’s monograph comprise a close exposition of convex integration concept for open and abundant kin with targeted proofs that have been frequently passed over in Gromov’s booklet. (…) Spring’s ebook makes no try and contain all subject matters from convex integration thought or to discover the entire gemstones in Gromov’s primary account, however it will still (or accurately accordingly) take its position as a typical reference for the speculation subsequent to Gromov’s towering monograph and will turn out critical for someone wishing to benefit in regards to the thought in a extra systematic means.

- Mathematical Reviews

This quantity offers a complete research of convex integration conception. (…) We suggested the ebook warmly to all drawn to differential topology, symplectic topology and optimum regulate theory.

- Matematica

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**Additional resources for Convex Integration Theory: Solutions to the h-principle in geometry and topology**

**Sample text**

12 to this initial C-structure (h0 , H0 ), one obtains a C-structure (h, H) over B which satisﬁes both of the properties (i), (ii). 14 (C ∞ -structures). Suppose p : E → B is a smooth aﬃne Rq bundle, and that f ∈ Γ(E), β ∈ Γ(R) are smooth sections. There is a C-structure (h, H) over B with respect to f, β such that H (and hence h) is a smooth map 1 and such that for all b ∈ B, f (b) = 0 h(t, b) dt. Proof. 7, there is a smooth C-structure (h0 , H0 ) over B with respect to f, β. 12 applied to this initial C-structure (h0 , H0 ), subject to the following modiﬁcations to ensure a smooth reparametrization.

21) it follows that for all > 0, over Op K, the image of F , hence also the image of ∂t f , lies in N (δ/2) In particular, over Op K, ∂t f − ∂t f0 = ∂t f − β < δ/2. 24) §1. THE ONE-DIMENSIONAL THEOREM 41 Since B is a C 1 -manifold there is a function λ ∈ C 1 (B, [0, 1]), compactly supported in Op K, such that λ = 1 in a neighbourhood W of K. Let f = f + λ (f0 − f ). Thus f is C 1 in t, f = f0 on W and f = f on B \ Op K. e. f satisﬁes the approximation property (i). Let F = F : [0, 1] → Γ(E) be the homotopy, F (s, b) = F (s, b) + s ∂t (λ (f0 − f ))(b).

F − f0 ≤ . In addition to the basic properties (i), (ii), the full development of the theory requires two additional properties: a homotopy property and a relative property (properties (iii), (iv) below). Both of these additional properties are required in subsequent chapters for the proof of the h-principle. Brieﬂy, the h-principle is proved inductively over a locally ﬁnite covering of the base manifold by charts. Thus, relative properties are essential in order to preserve what was established in previous steps of the induction.