By Abel Flint

**Read or Download A system of geometry and trigonometry: Together with a treatise on surveying : teaching various ways of taking the survey of a field, also to protract ... using them : compiled from various authors PDF**

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**Extra info for A system of geometry and trigonometry: Together with a treatise on surveying : teaching various ways of taking the survey of a field, also to protract ... using them : compiled from various authors**

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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.