Chapter 7 Partitions of Unity, Orientability, Covering Maps ~
... Theorem 7.4. Let M be a smooth manifold and let {U↵}↵2I be an open cover for M . Then, there is a countable partition of unity, {fi}i 1, subordinate to the cover {U↵}↵2I and the support, supp fi, of each fi is compact. If one does not require compact supports, then there is a partition of unity, {f↵ ...
... Theorem 7.4. Let M be a smooth manifold and let {U↵}↵2I be an open cover for M . Then, there is a countable partition of unity, {fi}i 1, subordinate to the cover {U↵}↵2I and the support, supp fi, of each fi is compact. If one does not require compact supports, then there is a partition of unity, {f↵ ...
Notes on von Neumann Algebras
... Theorem 3.2.1. Let M be a self-adjoint subalgebra of B(H) containing 1, with dim(H) = n < ∞. Then M = M 00 . Proof. It is tautological that M ⊆ M 00 . So we must show that if y ∈ M 00 then y ∈ M . To this end we will “amplify” the action of M on H to an action on H⊗H defined by x(ξ⊗η) = xξ⊗η. If we ...
... Theorem 3.2.1. Let M be a self-adjoint subalgebra of B(H) containing 1, with dim(H) = n < ∞. Then M = M 00 . Proof. It is tautological that M ⊆ M 00 . So we must show that if y ∈ M 00 then y ∈ M . To this end we will “amplify” the action of M on H to an action on H⊗H defined by x(ξ⊗η) = xξ⊗η. If we ...
AN INTRODUCTION TO (∞,n)-CATEGORIES, FULLY EXTENDED
... completely determined by the image of S 1 together with its algebraic structure. Indeed, it is possible to see that a 2-dimensional TQFT is equivalent to the datum of a commutative Frobenius algebra structure on a finite-dimensional vector space A, which is precisely F (S 1 ) - see [Koc04] for a pro ...
... completely determined by the image of S 1 together with its algebraic structure. Indeed, it is possible to see that a 2-dimensional TQFT is equivalent to the datum of a commutative Frobenius algebra structure on a finite-dimensional vector space A, which is precisely F (S 1 ) - see [Koc04] for a pro ...
Graded Brauer groups and K-theory with local coefficients
... Our aim is to define a (c K-theory with local coefficients 5? K^X) (K denotes either KO or KU) which shall generalize the usual groups K^X), yzeZg or TzeZg. The ordinary cohomology with local coefficients H^X, a) is defined for (n, o^eZxH^X.Z^). At least when X is a connected finite CW-complex, KO^X ...
... Our aim is to define a (c K-theory with local coefficients 5? K^X) (K denotes either KO or KU) which shall generalize the usual groups K^X), yzeZg or TzeZg. The ordinary cohomology with local coefficients H^X, a) is defined for (n, o^eZxH^X.Z^). At least when X is a connected finite CW-complex, KO^X ...
