By David Spring

This booklet offers a complete research of convex integration conception in immersion-theoretic topology. Convex integration conception, constructed initially by way of M. Gromov, presents basic topological tools for fixing the *h*-principle for a wide selection of difficulties in differential geometry and topology, with functions additionally to PDE conception and to optimum keep an eye on conception. even though topological in nature, the idea is predicated on an actual analytical approximation consequence for greater order derivatives of services, proved by way of M. Gromov. This booklet is the 1st to offer an exacting checklist and exposition of all the uncomplicated strategies and technical result of convex integration idea in better order jet areas, together with the idea of iterated convex hull extensions and the speculation of relative *h*-principles. A moment function of the publication is its distinct presentation of functions of the overall idea to issues in symplectic topology, divergence unfastened vector fields on 3-manifolds, isometric immersions, absolutely genuine embeddings, underdetermined non-linear structures of PDEs, the comfort theorem in optimum regulate concept, in addition to purposes to the normal immersion-theoretical issues equivalent to immersions, submersions, *k*-mersions and unfastened maps.

The booklet may still turn out invaluable to graduate scholars and to researchers in topology, PDE conception and optimum keep watch over thought who desire to comprehend the *h*-principle and the way it may be utilized to resolve difficulties of their respective disciplines.

------ studies

The first 8 chapters of Spring’s monograph comprise a close exposition of convex integration concept for open and plentiful kinfolk with precise proofs that have been usually passed over in Gromov’s booklet. (…) Spring’s publication makes no try and contain all themes from convex integration idea or to discover all the gemstones in Gromov’s primary account, however it will still (or accurately accordingly) take its position as a customary reference for the speculation subsequent to Gromov’s towering monograph and will end up fundamental for an individual wishing to benefit concerning the conception in a extra systematic manner.

- Mathematical Reviews

This quantity presents a accomplished examine of convex integration concept. (…) We instructed the booklet warmly to all drawn to differential topology, symplectic topology and optimum keep an eye on theory.

- Matematica

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

**Sample text**

We identify Γr (XU ) ≡ C r (U, Rq ), 0 ≤ r ≤ ∞. The manifold X (1) is represented locally as, n (1) XU = U ×W × Rq = J 1 (U, W ). 1) 1 where W ⊂ Rq is open. With respect to local coordinates (u1 , u2 , . .

The s,r q s purpose of the space C (B, R ) is to allow for C -approximations, s large, in the U -coordinates of split manifold B = I n−1 × [0, 1]. 6. e. the norm r,r coincides with the usual sup-norm for the C r -topology on the function space C r (B, Rq ). For all k ≥ s, the inclusion i : C k (B, Rq ) → C s,r (B, Rq ) is continuous. 8 below is as follows: Let T : C s (B, Rq ) → C s (B, Rq ) be the map, t T (f )(u, t) = f (u, s) ds. 0 Then, T : C s,r (B, Rq ) → C s,r+1 (B, Rq ) is continuous (s ≥ r + 1).

E. the perturbed section f also satisﬁes conclusion (i). §2. 7 to the One-Dimensional Theorem is generalized to the C ⊥ -approximation theorem, which is the main technical result that is employed in subsequent chapters for proving the h-principle. 7 proves C s -approximations in the corresponding U -derivatives with respect to coordinates in I n−1 . The generalization consists of extending these approximations to include the pure derivatives ∂tj , 0 ≤ j ≤ r − 1. 8. Let π : E = B × Rq → B be the product Rq -bundle over the n-cube B = [0, 1]n in Rn , n ≥ 1, and let R ⊂ E be open.