×
Crabb, Michael; Ranicki, Andrew

The geometric Hopf invariant and double points. (English) Zbl 1205.55011

J. Fixed Point Theory Appl. 7, No. 2, 325-350 (2010).
Summary: The geometric Hopf invariant of a stable map \(F\) is a stable \({\mathbb Z}/2\) -equivariant map \(h(F)\) such that the stable \({\mathbb Z}/2\) -equivariant homotopy class of \(h(F)\) is the primary obstruction to \(F\) being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map \(F\) of an immersion \(f: M^m \looparrowright N^n\) in terms of the double point set of \(f\). We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions \(f\) in the metastable dimension range \(3m < 2n - 1\) (when a generic \(f\) has no triple points) in terms of the geometric Hopf invariant.

Cited in 1 Document

MSC:

55Q25 Hopf invariants
57R42 Immersions in differential topology

Keywords:

geometric Hopf invariant; immersion; double point
Cite Review PDF
Full Text: DOI arXiv

References:

[1] J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture. In: Proc. 1982 Aarhus Topology Conf., Springer Lecture Notes 1051, 1984, 483–532. · Zbl 0553.55010
[2] Boardman M., Steer B.: Hopf invariants. Comment. Math. Helv. 42, 217–224 (1967) · Zbl 0166.19201 · doi:10.1007/BF02564417
[3] M. C. Crabb, \({{\mathbb Z}/2}\) -Homotopy Theory. LMS Lecture Notes 44, Cambridge University Press, Cambridge, 1980.
[4] Crabb M.C.: The homotopy coincidence index. J. Fixed Point Theory Appl. 7, 1–32 (2010) · Zbl 1205.55002 · doi:10.1007/s11784-010-0006-z
[5] Crabb M.C., James I.M.: Fibrewise Homotopy Theory. Springer, Berlin (1998) · Zbl 0905.55001
[6] M. C. Crabb and A. A. Ranicki, The geometric Hopf invariant and surgery theory. In preparation. Preliminary announcement available at http://www.maths.ed.ac.uk/aar/slides/geohopf.pdf . · Zbl 1404.55001
[7] Dax J.P.: Étude homotopique des espaces de plongements. Ann. Sci. École Norm. Sup. (4) 5, 303–377 (1972) · Zbl 0251.58003
[8] A. Haefliger, Plongements différentiables dans le domaine stable. Comment. Math. Helv. 37 (1962/1963), 155–176. · Zbl 0186.27302
[9] Haefliger A., Hirsch M.W.: Immersions in the stable range. Ann. of Math. (2) 75, 231–241 (1962) · Zbl 0186.27301 · doi:10.2307/1970171
[10] Hatcher A., Quinn F.: Bordism invariants of intersections of submanifolds. Trans. Amer. Math. Soc. 200, 327–344 (1974) · Zbl 0291.57019 · doi:10.1090/S0002-9947-1974-0353322-6
[11] Hirsch M.W.: Immersions of manifolds. Trans. Amer. Math. Soc. 93, 242–276 (1959) · Zbl 0113.17202 · doi:10.1090/S0002-9947-1959-0119214-4
[12] Klein J. R.: Poincaré complex diagonals. Math. Z. 258, 587–607 (2008) · Zbl 1137.57024 · doi:10.1007/s00209-007-0187-3
[13] Klein J.R., Williams E.B.: Homotopical intersection theory. I Geom. Topol. 11, 939–977 (2007) · Zbl 1132.57024
[14] Klein J.R., Williams E.B.: Homotopical intersection theory. II. Equivariance. Math. Z. 264, 849–880 (2010) · Zbl 1185.57031 · doi:10.1007/s00209-009-0491-1
[15] U. Koschorke, Vector Fields and Other Vector Bundle Morphisms–A Singularity Approach. Lecture Notes in Mathematics 847, Springer, Berlin, 1981. · Zbl 0459.57016
[16] Koschorke U.: Nielsen coincidence theory in arbitrary codimensions. J. reine angew. Math 598, 211–236 (2006) · Zbl 1107.55002 · doi:10.1515/CRELLE.2006.075
[17] Koschorke U., Sanderson B.: Geometric interpretations of the generalized Hopf invariant. Math. Scand. 41, 199–217 (1977) · Zbl 0385.55006
[18] Li B., Liu R., Zhang P.: A simplified approach to embedding problem in normal bordism framework. Systems Sci. Math. Sci. 5, 180–192 (1992) · Zbl 0763.55002
[19] Ranicki A.A.: The algebraic theory of surgery II. Applications Proc. Lond. Math. Soc. (3) 40, 193–283 (1980) · Zbl 0471.57011 · doi:10.1112/plms/s3-40.2.193
[20] Salomonsen H.A.: Bordism and geometric dimension. Math. Scand. 32, 87–111 (1973) · Zbl 0264.57009
[21] Smale S.: Regular curves on Riemannian manifolds. Trans. Amer. Math. Soc. 87, 492–512 (1958) · Zbl 0081.38103 · doi:10.1090/S0002-9947-1958-0094807-0
[22] Smale S.: The classification of immersions of spheres in Euclidean spaces. Ann. of Math. (2) 69, 327–344 (1959) · Zbl 0089.18201 · doi:10.2307/1970186
[23] C. T. C. Wall, Surgery on Compact Manifolds. 2nd ed., AMS, Providence, RI, 1999. · Zbl 0935.57003
[24] Whitney H.: The self-intersections of a smooth n-manifold in 2n-space. Ann. of Math. (2) 45, 220–246 (1944) · Zbl 0063.08237 · doi:10.2307/1969265
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.