The proof took more than 9 months to be found but is now quite clear: the key is a multiplicative analogue of the Poincaré-Hopf theorem telling what happens if a cell is added to the complex: we just multiply with the Poincaré-Hopf index which is , where S(x) is the the unit sphere of the new cell, the boundary of the cell which got attached ...
Jun 19, 2018 · The result generalizes a result of Hassler Whitney from 1931, who proved a theorem implying that all 4-connected maximal planar graphs are Hamiltonian. But 4-connected maximal planar graphs are exactly the class of 2-spheres. Whitney’s result was later generalized by Bill Tutte and generalized to 4-connected planar graphs.
A discrete analogue of the classic Whitney–Graustein theorem is proven by showing that the winding number of polygons is a complete invariant for this classification. Moreover, this proof is constructive in that for any pair of equivalent polygons, it produces some sequence of regular transformations taking one polygon to the other.
Whitney para funciones de clase C1;1 en espacios de Hilbert, obteniendo además un control óptimo de la constante Lipschitz del gradiente de la extensión, en términos del 1-jet inicial. La demostración de Wells
A contact geometric proof of the Whitney-Graustein theorem. The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.
Proof that immersions and submersions are open subsets in the Whitney topology. 1/10/08: Proof that embeddings form an open set in the Whitney topology.
Proof Assume (2.4) holds. By Definition 1.5, for any X,Xo,Xl.f, k(x,xo)<k(x,xl)+k(xl,xo). Let " be a Whitney cover of f consisting of cubes Qj with centers xy. Then for any xy, due to Minkowski’sinequalityandanelementaryinequality, Itl <,lt,lr, where0<r<1, andLemma1.13, wehave (fflk(x,xo)Sd#) _ k(x,xj)’d# + k(xj,xo)Sd# < (k(x,xy))Sd# < (k(x,xy))Sd# +[QfQk(xy’x)Sd#] (l/s)
The Residue Theorem and Applications - Vaughan McDonald. 2017 Counselor Seminars on Characteristic Classes. Vector Bundles - Kevin Lin. Stiefel-Whitney Classes - Kevin Lin. The Euler Class and the Thom Isomorphism - Kevin Lin. Chern Classes - Kevin Lin. The Chern-Weil Homomorphism - Kevin Lin. 2017 Counselor Seminars on Category Theory
Jan 28, 2020 · Proof. By the argument in the proof of prop. , the only possible obstruction is the second Stiefel-Whitney class w 2 w_2.By the discussion at Wu class, this vanishes on an oriented manifold precisely if the second Wu class vanishes.