By Stein M.R., Dennis R.K. (eds.)

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**Extra resources for Algebraic K-Theory and Algebraic Number Theory**

**Example text**

Given a compact domain R ~ P with smooth boundary 8R, let Ql, ... , Qk be the connected components of 8R. Suppose each

We can therefore define a new maximal atlas, A max , associated with the given smooth structure: A max = {f: V -+ V' I V ~ M n open, V' ~ IR n open, f diffeomorphism}. The inverse diffeomorphisms f-1: V' -+ V will be called local parametrizations. 2 it follows that every point in a smooth manifold M n has an open neighborhood V ~ M n that is diffeomorphic to IR n . e define an equivalence From now on chart will mean a chart in the maximal atlas. Alternatively one can the canonical projection )n.

11 Let M n be a smooth manifold of dimension n. There exists an embedding of M n into a Euclidean space IRn+k. r,J. ";t 8. , fd(q), 1>l(q), ... , 1>d(q))ยท :0 V) and such that . surface when n = 2). 8. 11 can her hand k cannot be Assuming f(ql) = f(Q2), we can by (i) choose j such that ql E Uj. Then 1>j(q2) = 1>j(Ql) =I- 0, Q2 E Uj, and by (ii), ql = Q2. Hence f is injective. Since M is compact, f is a homeomorphism from M to f(M). Let 7fl: U{ ---+ IR n(d-l)+d ,. Define a diffeomorphism hI from dijJeomorphically onto rna A.