Boy's surface

Abstract: An analysis of Boy's immersion of the projective plane in 3-space is given via a collection of planar figures. An analogous construction yields an immersion of the 3-sphere in 4-space which represents a generator of the third stable stem. This immersion has one quadruple point and a closed curve of triple points whose normal matrix is a 3-cycle. Thus the corresponding multiple point invariants do not vanish. The construction is given by way of a family of three dimensional cross sections. Stable homotopy groups can be interpreted as bordism groups of immersions. The isomorphism between such groups is given by the Pontryagin-Thom construction on the normal bundle of representative immersions. The self-intersection sets of immersed n-manifolds in Rn + 1 provide stable homotopy invariants. For example, Boy's immersion of the projective plane in 3-space represents a generator of 1'(P?); this can be seen by examining the double point set. In this paper a direct analog of Boy's surface is constructed. This is an immersion of S3 in 4-space which represents a generator of 11; the self-intersection data of this immersion are as simple as possible. It may be that appropriate generalizations of this construction will exemplify a nontrivial Kervaire invariant in dimension 30, 62, and so forth. I. Background. Let i: Mn ,+ RW ' denote a general position immersion of a closed n-manifold into real (n + 1)-space. Assume that i and M are smooth, and that Rn+" is given the standard differential structure. By applying the Pontryagin-Thom construction on the normal bundle of (i, M), one obtains a stable homotopy class in nL1+l(P). Here P' denotes the infinite real projective space, an Eilenberg-Mac Lane space of type K(Z/2, 1). (L1js(X) = limk[Sk+j, SkX], for X a locally compact Hausdorff space; Sk(_) is the reduced suspension.) If the pair (i, M) is varied by a bordism, the same homotopy class arises. Furthermore, if M is oriented, then the corresponding homotopy class lies in LIs (= FIIs I(S') by adjointness). Finally any class in fI7JS+? (P ?) or Hls may be realized in this way. These isomorphisms were noticed first by Wells [20], see also [16] for a modern proof. Stable homotopy has recently been approached from the immersion point of view (see [5, 7, 14, and 19] for example). Since immersions are concrete geometric objects, this new approach brings into focus many of the terse ideas of homotopy theory. One can distinguish immersions by means of their self-intersection data; e.g. the Received by the editors March 9, 1984. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R42, 57R65, 55Q45, 57N35. ?1986 American Mathematical Society 0002-9947/86 $1.00 I $.25 per page 103 This content downloaded from 157.55.39.213 on Sat, 02 Jul 2016 05:46:31 UTC All use subject to http://about.jstor.org/terms

Abstract: We consider C ∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C ∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.

Abstract: Boy’s surface is the simplest and most symmetrical way of making a compact model of the projective plane in R 3 without any singular points. This surface has 3-fold rotational symmetry and a single triple point from which three loops of intersection lines emerge. It turns out that there is a second, homeomorphically different way to model the projective plane with the same set of intersection lines, though it is less symmetrical. There seems to be only one such other structure beside Boy’s surface, and it thus has been named Girl’s surface. This alternative, finite, smooth model of the projective plane seems to be virtually unknown, and the purpose of this paper is to introduce it and make it understandable to a much wider audience. To do so, we will focus on the construction of the most symmetrical Mobius band with a circular boundary and with an internal surface patch with the intersection line structure specified above. This geometry defines a Girl’s cap with C2 front-to-back symmetry.

Abstract: We show that the symmetry group of a stable immersion of the real projective plane P in E^3 is either trivial or is cyclic of order 3, and that of a stable map of P in E^3 is conjugate to a subgroup of the full tetrahedral group. Thus Boy's surface, in its `standard' form, is the most symmetrical stable immersion of P in $E^3, and Steiner's surface is given by the most symmetrical stable map of P in E^3. We also construct a smooth embedding of P in E^4 with symmetry group SO(2) by orthogonal projection of the Veronese surface.