By Gunnar E. Carlsson, Ralph L. Cohen, Wu-Chung Hsiang, John D.S. Jones

In 1989-90 the Mathematical Sciences learn Institute performed a application on Algebraic Topology and its purposes. the most parts of focus have been homotopy conception, K-theory, and functions to geometric topology, gauge idea, and moduli areas. Workshops have been performed in those 3 parts. This quantity contains invited, expository articles at the issues studied in this application. They describe fresh advances and aspect to attainable new instructions. they need to turn out to be worthwhile references for researchers in Algebraic Topology and similar fields, in addition to to graduate scholars.

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**Example text**

W4 . Now suppose M is a manifold of dimension n, and P a G-structure on M . 1. Thus ρ1 (P ), . . , ρ4 (P ) are vector bundles over M . Clearly, ρ2 (P ) is a vector subbundle of ρ1 (P ), and the quotient bundle ρ1 (P )/ρ2 (P ) is ρ3 (P ). If ∇ is any connection on P , then its torsion T (∇) lies in C ∞ (ρ1 (P )), and if ∇, ∇ are two connections on P , then T (∇ ) − T (∇) lies in the subspace C ∞ (ρ2 (P )) of C ∞ (ρ1 (P )). Therefore, the projections of T (∇) and T (∇ ) to the quotient bundle ρ3 (P ) = ρ1 (P )/ρ2 (P ) are equal.

E. if d E ∇γ(t) s(t) = 0 for all t ∈ [0, 1], where γ(t) ˙ is dt γ(t), regarded as a vector in Tγ(t) M . ˙ Now this is a ﬁrst-order ordinary differential equation in s(t), and so for each possible initial value e ∈ Eγ(0) , there exists a unique, smooth solution s with s(0) = e. We shall use this to deﬁne the idea of parallel transport along γ. 1 Let M be a manifold, E a vector bundle over M , and ∇E a connection on E. Suppose γ : [0, 1] → M is smooth, with γ(0) = x and γ(1) = y, where x, y ∈ M .

Suppose ex ∈ Ex , and Pα (ex ) = ey ∈ Ey . Then there is a unique parallel section s of α−1 (E) with s(0) = ex and s(1) = ey . Deﬁne s (t) = s(1 − t). Then s is a parallel section of (α−1 )∗ (E). Since s (0) = ey and s (1) = ex , it follows that Pα−1 (ey ) = ex . Thus, if Pα (ex ) = ey , then Pα−1 (ey ) = ex , and so Pα and Pα−1 are inverse maps. In particular, this implies that if γ is any piecewise-smooth path in M , then Pγ is invertible. By a similar argument, we can also show that Pβα = Pβ ◦ Pα .