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By Abel Flint

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Complex Numbers and Geometry (MAA Spectrum Series)

The aim of this publication is to illustrate that complicated numbers and geometry could be mixed jointly superbly. This leads to effortless proofs and typical generalizations of many theorems in aircraft geometry, corresponding to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.

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1): that X be a simple Poincar´e complex. A second such condition concerns ν: if the induced bundle over M agrees (stably) with the normal bundle of an embedding in Euclidean space, then it follows by a standard application of the Thom-Pontrjagin construction (shrinking the complement of a tubular neighbourhood to a point) that the Thom space of ν is (stably) reducible. Thus ν must be stably fibre homotopy equivalent to the ‘Spivak normal fibration’ of the Poincar´e complex X, as described in [S5] and [W4].

3. statement of results 37 We now state the geometrical part of our results. 2. Let X be a simple Poincar´e (n + 1)-ad, M a compact manifold (n + 1)-ad, φ : M → X a map of degree 1; ν a bundle over |X|, F a stable trivialisation of τ|M| ⊕ φ∗ ν. Then there is an obstruction θ in Ldim |M| π(X) which depends only on the bordism class of (M, φ, F ) and vanishes if that class contains a simple homotopy equivalence; conversely, if dim M { } > 4, its vanishing is sufficient. If we suppose also that φ induces a simple homotopy equivalence ∂j M → ∂j X, and keep ∂j M fixed in the surgeries, then the appropriate obstruction lies in Ldim |M| δj π(X) , and its vanishing suffices if dim M { } 4.

The exact sequence π2 (P ) → H2 (P ) → H2 (G) = 0 shows that we can pick a base of H2 (P ), represent each element by a map of a 2sphere, and attach corresponding 3-cells. This gives a complex L with H2 (L) = H3 (L) = 0 and still π1 (L) ∼ = G. If we now attach a 2-cell by α, we obtain a simply connected complex K with no homology; hence K is contractible. Now apply the theorem with r = 3, m 5; then ψ is at least 2-connected, so induces an isomorphism of fundamental groups. The result follows. Corollary.