Abstract
The applications to surgery obstruction theory. A brief review of geometric Poincaré complexes, normal maps and Umkehr maps. The geometric Hopf invariant of the Umkehr of a normal map gives the surgery obstruction of a normal map, both in the simply-connected case and in general, as well as for the ultranormal maps arising in codimension 2 surgery theory (the Seifert forms of knots). The spectral quadratic construction is applied to give the total surgery obstruction of a geometric Poincaré complex.
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Notes
- 1.
Using twisted coefficients in the nonorientable case.
- 2.
A differentiable manifold is triangulable and is a CW complex, whereas a topological manifold need not be triangulable and only has the homotopy type of a CW complex.
- 3.
Which applies because the Spivak normal fibration of X has a vector bundle reduction.
- 4.
There is no smooth analogue of the total surgery obstruction, and \(\mathcal {S}^O(X)\) does not have the structure of an abelian group, by Crowley [17].
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Crabb, M., Ranicki, A. (2017). Surgery Obstruction Theory. In: The Geometric Hopf Invariant and Surgery Theory. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-71306-9_8
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DOI: https://doi.org/10.1007/978-3-319-71306-9_8
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-71306-9
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