Self-intersections and higher Hopf invariants

Koschorke, Ulrich; Sanderson, Brian

doi:10.1016/0040-9383(78)90032-0

Cited by 46 publications

(44 citation statements)

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“…The key idea enabling this to be done is the geometric formulation of stable Hopf invariants given by U.Koschorke and B. J. Sanderson (17). A more detailed discussion of the ideas which follow in the case of oriented codimension one immersions may be found in ((11); § 2).…”

Section: Reducing the Problem To Homology Theorymentioning

confidence: 99%

Codimension one immersions and the Kervaire invariant one problem

Eccles

1981

Math. Proc. Camb. Phil. Soc.

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IntroductionLet i : M<\ > U n+1 be a self-transverse immersion of a compact closed smootĥ -dimensional manifold in (n+ 1)-dimensional Euclidean space. A point of U n+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self^transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+l-r (the empty set if r > n+ 1). In particular, the set of (n+ l)-fold intersection points is finite of order, say, 6(i). In this paper we are concerned with the set of values of 6(i) for (selftransverse) immersions of all (compact closed smooth) manifolds of given dimension n.Placing n +1 copies of the ^-sphere S n in general position in U n+1 provides an immersion i 0 : \jS nc \ > U n+1 with 6(i 0 ) = 2. Thus given an immersion i x : M3 > U n+1 disjoint union with i 0 provides another immersion i % with 6(i 2 ) = 0(ij) + 2. On the other hand, given an immersion i x : M9 > U n+1 with 0(i x ) ^ 2 handles may be attached to M to eliminate two of the (n + 1 )-fold points giving an immersion i 2 with 6(i 2 ) = d(i x ) -2 (the argument for n = 2 is given in detail by T. F. Banchoff (2)). This leaves the problem of whether d(i) can be odd.In a previous paper (11) it was shown that if M is orientable then 6(i) can be odd if and only if n = 0,1 or 3 (see also (18)). The method of proof was to observe that the parity of 6 is a bordism invariant and so to translate the problem into homotopy theory where it could be solved using standard techniques of algebraic topology. The same method can be applied in the general case, as explained in § 2, but so far has not led to a complete solution. However it does show that for certain values of n the problem is equivalent to well known problems in homotopy theory. The main result of this paper is as follows.

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Section: Reducing the Problem To Homology Theorymentioning

confidence: 99%

Codimension one immersions and the Kervaire invariant one problem

Eccles

1981

Math. Proc. Camb. Phil. Soc.

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“…(The proof that g is an equivalence in [36, 4.8] is not quite right; it is corrected in [14, p. 485] and also in [30]. )…”

Section: (L)~) ~ (L -{Y}x Il-{y}) If X(y) = "~ •mentioning

confidence: 99%

“…Sanderson [30], who discovered them independently of Cohen and Taylor. We digress to mention an application to Thorn spectra in [18].…”

Section: J Qmentioning

confidence: 99%

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Applications and generalizations of the approximation theorem

May¹

1979

Lecture Notes in Mathematics

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“…This section is finished by proving that ck are the Chern classes using the computations of the homology of infinite loop spaces in [3,6 and 11]. In the third section, if /¿: M -> BU is a classifying map of the bundle £, we associate an immersion /: N -> M to the composition t/^: M -> QBU(\) following [7]. Then, we prove that ck(i-) is the Poincaré dual of the fundamental class of the k-tuple points manifold of the immersion/.…”

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confidence: 99%

A geometric interpretation of the Chern classes

Villanueva¹

1983

Trans. Amer. Math. Soc.

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Abstract. Letff M -» BU be a classifying map of the stable complex bundle £ over the weakly complex manifold M. If t is the stable right homotopical inverse of the infinite loop spaces map tj: QBU(\) -» BU, we define/£' = t /{ and we prove that the Chern classes ck(£) are/j'*(A*(ia)), where hk is given by the stable splitting of QBU(\) and tk is the Thorn, class of the bundle y'*' = E1kJf2 yk. Also, we associate to/' an immersion g: N -> M and we prove that ck(í) is the dual of the image of the fundamental class of the £-tuple points manifold of the immersion g, g*([Nk]).1. Introduction. In this paper, we give a geometric interpretation of the Chern classes of a stable complex vector bundle £ over a weakly complex manifold M. In the second section we define characteristic classes ck in H2k(BU; Z) using the weak homotopy equivalence between QBU(\) and CBU(\) for an appropriate coefficient system Q got in [9,4], the stable splitting of CX given in [4] and the stable map t:

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Self-intersections and higher Hopf invariants

Cited by 46 publications

References 9 publications

Codimension one immersions and the Kervaire invariant one problem

Codimension one immersions and the Kervaire invariant one problem

Applications and generalizations of the approximation theorem

A geometric interpretation of the Chern classes

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