By Bass

Complaints Of The convention Held on the Seattle study heart Of Battelle Memorial Institute, August 28 - September eight, 1972

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**Extra resources for Algebraic K-Theory III**

**Sample text**

There is a natural homomorphism K(X)→K(X) sending E − εn to the ∼ class of E . This is well-defined since if E − εn = E ′ − εm in K(X) , then E ⊕ εm ≈s E ′ ⊕ εn , hence E ∼ E ′ . The map K(X)→K(X) is obviously surjective, and its kernel consists of elements E −εn with E ∼ ε0 , hence E ≈s εm for some m , so the kernel consists of the elements of the form εm − εn . This subgroup {εm − εn } of K(X) is isomorphic to Z . In fact, restriction of vector bundles to a basepoint x0 ∈ X defines a homomorphism K(X)→K(x0 ) ≈ Z which restricts to an isomorphism on the subgroup {εm − εn } .

We claim that E(σ ) is homeomorphic to a closed ball. To prove this the main step is to show that the projection π : E(σ )→H1 , π (v1 , · · · , vn ) = v1 , is a trivial fiber bundle. This is equivalent to finding a projection p : E(σ )→π −1 (v0 ) which is a homeomorphism on fibers of π , where v0 = (0, · · · , 0, 1) ∈ Rσ1 ⊂ Rk , since the map π × p : E(σ )→H1 × π −1 (v0 ) is then a continuous bijection of compact Hausdorff spaces, hence a homeomorphism. The map p : π −1 (v)→π −1 (v0 ) is obtained by applying the rotation ρv of Rk that takes v to v0 and fixes the (k − 2) dimensional subspace orthogonal to v and v0 .

For Uℓ to be open is equivalent to its preimage in Vn (Rk ) being open. This preimage consists of orthonormal frames v1 , · · · , vn such that πℓ (v1 ), · · · , πℓ (vn ) are independent. Let A be the matrix of πℓ with respect to the standard basis in the domain Rk and any fixed basis in the range ℓ . The condition on v1 , · · · , vn is then that the n× n matrix with columns Av1 , · · · , Avn have nonzero determinant. Since the value of this determinant is obviously a continuous function of v1 , · · · , vn , it follows that the frames v1 , · · · , vn yielding a nonzero determinant form an open set in Vn (Rk ) .