We completely characterize real Bott manifolds up to diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with Z/2 coefficients is induced by an affine diffeomorphism betwe...

A real Bott manifold is the total space of a sequence of RP 1 bundles starting with a point, where each RP 1 bundle is the projectivization of a Whitney sum of two real line bundles. A real Bott manifold is a real toric manifold which admits a flat riemannian metric. An upper triangular (0, 1) matrix with zero diagonal entries uniquely determines such a sequence of RP 1 bundles but different ma...

Part 1. Vector Bundles and Bott Periodicity 2 1. Families of Vector Spaces and Vector Bundles: 8/27/15 2 2. Homotopies of Vector Bundles: 9/1/15 5 3. Abelian Group Completions and K(X): 9/3/15 8 4. Bott’s Theorem: 9/8/15 12 5. The K-theory of X × S2: 9/10/15 15 6. The K-theory of the Spheres: 9/15/15 18 7. Division Algebras Over R: 9/17/15 22 8. The Splitting Principle: 9/22/15 25 9. Flag Manif...

Abstract. A real Bott manifold is the total space of iterated RP 1 bundles starting with a point, where each RP 1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant u...

This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim at completeness. For instance, the variants of Bott periodicity related to bivariant K-theory are described by Cuntz in this handbook. As another example, we don’t emphasize here the relation between motivic homotopy theory and Bott periodicit...

Let A be a bounded linear operator, PM be an orthogonal projection with range M and PM ,N be an idempotent with range M and kernel N . This paper presents some novel relations between Bott-Duffin inverse AM = PM (APM + PM⊥) + and generalized Bott-Duffin inverse AM ,N = PM ,N (APM ,N + PN ,M ) + . Furthermore, the representations for the BottDuffin inverse and generalized Bott-Duffin inverse are...

Figure 1. Raoul Bott in 2002. Raoul Bott passed away on December 20, 2005. Over a five-decade career he made many profound and fundamental contributions to geometry and topology. This is the second part of a two-part article in the Notices to commemorate his life and work. The first part was an authorized biography, “The life and works of Raoul Bott” [4], which he read and approved a few years ...

1 Description The Periodicity Theorem was proved by Raoul Bott over fifty years ago (cf. survey [3], [4], [9]) and quickly became one of the strongest tools in homotopy theory, topology of manifolds and global analysis. The original theorem asserted that homotopy groups of the linear groups GL(n,F) where F is the field of real, complex or quaternion numbers are periodic i.e. πi(GL(k,F) ' πi+nF(...

There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-CattaneoRossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown...

A quasitoric manifold (resp. a small cover) is a 2ndimensional (resp. an n-dimensional) smooth closed manifold with an effective locally standard action of (S) (resp. (Z2)n) whose orbit space is combinatorially an n-dimensional simple convex polytope P . In this paper we study them when P is a product of simplices. A generalized Bott tower over F, where F = C or R, is a sequence of projective b...