Topology of Sobolev mappings, II

TY - JOUR

T1 - Topology of Sobolev mappings, II

AU - Hang, Fengbo

AU - Lin, Fanghua

N1 - Funding Information: Our proof is somewhat different from the one in \[B2\]. This modification becomes necessary because we have troubles with the original proof, given in \[B2\], with regard to matching the boundary values when patching cubes for the case n-p> 1. Moreover, in studying the problem whether a specific map can be approximated in the strong topology by a sequence of smooth maps, we need the explicit Construction in our proof of Theorem 6.1. As a consequence we know that for 1 ~<p < n, if p ~ Z or p--1 or 2 ~<p < n but peZ and 7rp(N)=0, then H~'P(M,N )=H~P(M, N) (see \[B2\], Theorem 7.2 and \[Hn\]). The case 2~p<n, pEZ and 7rp(N)r is much more subtle. On the other hand, we have (see Theorem 7.2), for l~<p<n, H~'B(M,N )=WI'P(M, N) if and only if 7r\[p\](N)=0a nd H~P(M, N)=WI,B(M, N). Our proof of Theorem 6.1 also relies on various analytical estimates, some of which were obtained in the earlier work of Bethuel \[B2\]. The proof of the main theorem in w (Theorem 6.3) uses in a crucial way certain new deformations from the so-called dual skeletons, which is obviously motivated by the well-known work of Federer and Fleming on normal and integral currents (see \[Fe\],i n particular Chapter 4). The construction of such deformations with the right analytical estimates is the key point of the whole proof. We note that the previously constructed deformations due to B. White \[Whl\] (or that in \[Hj\])d o not seem to work for our purpose. Finally in w we discuss weak sequential density of smooth maps in Sobolev spaces. Several technical estimates concerning generic slices of Sobolev maps as well as estimates relative to the deformations constructed in w are included in the appendices. The present paper treats only compact manifolds without boundary. Essentially all the results discussed here can be generalized to the case that M has a smooth nonempty boundary OM. We shall return to these in a future article. Acknowledgement. Both authors wish to thank S. Cappell and F. Bogomolov for valuable discussions and suggestions concerning the obstruction theory and counterexamples in Corollary 5.5 and Remark 5.2. The second author also wishes to thank H. Brezis and Y. Li for sending him the preprint \[BL\]a nd for their interesting lectures. The research of the first author is supported by a Dean's Dissertation Fellowship of New York University. The research of the second author is supported by an NSF grant.

PY - 2003

Y1 - 2003

UR - http://www.scopus.com/inward/record.url?scp=0346897889&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346897889&partnerID=8YFLogxK

U2 - 10.1007/BF02392696

DO - 10.1007/BF02392696

M3 - Article

AN - SCOPUS:0346897889

SN - 0001-5962

VL - 191

SP - 55

EP - 107

JO - Acta Mathematica

JF - Acta Mathematica

IS - 1

ER -