A FAKE SMOOTH CP^{2}#RP^{4}

DANIEL RUBERMAN AND RONALD J. STERN

Abstract. We show that the manifold ∗RP^{4}# ∗ CP^{2}, which is homotopy equivalent but not homeomorphic to RP^{4}#CP^{2}, is in fact smoothable.

1. Introduction

In Kirby’s problem list [Kir97, Problem 4.82] and in a recent lecture at MSRI, P. Teichner raised the question of the smoothability of a certain non-orientable 4-manifold. In this note, we show that the manifold in question, denoted ∗RP^{4}# ∗ CP^{2}, which is homotopy equivalent but not homeomorphic to RP^{4}#CP^{2}, is in fact smoothable. The smooth model we construct will have the additional property that its universal cover is diffeomorphic to CP^{2}#\overline{CP}^{2}. To describe the manifold in question, we remind the reader that one of the first consequences of Freedman’s simply-connected surgery theory was a construction of a manifold ∗CP^{2}, sometimes called CH in honor of Chern, which is homotopy equivalent but not homeomorphic to CP^{2}. The manifold ∗CP^{2} is not smoothable for classical reasons: it has non-trivial Kirby-Siebenmann invariant KS ∈ Z_{2}. Given any simply-connected non-spin manifold M, a similar construction produces a homotopy equivalent ‘∗–partner’ ∗M with opposite Kirby-Siebenmann invariant [Teich96]. In 1983, the first author [Rub84] constructed what is in effect the ∗-partner of RP^{4}. The connected sum ∗CP^{2}# ∗ RP^{4} has trivial KSinvariant and so might expected to be smoothable; on the other hand [HKT94] it is not homeomorphic to CP^{2}#RP^{4}.

Theorem 1. The manifold ∗CP^{2}# ∗ RP^{4} has a smooth structure. Moreover, it has a smooth structure such that its universal cover is diffeomorphic to CP^{2}#\overline{CP}^{2}

The classification [HKT94] of non–orientable manifolds with π_{1} = Z_{2} implies that such manifolds which have b_{2} > 1 are smoothable if and only if their Kirby-Siebenmann invariant vanishes. Together with theorem 1 this yields:

Corollary 2. Let X be a closed non–orientable 4-manifold with π_{1}(X) = Z_{2}. Then X has a smooth structure if and only if KS(X) = 0.

The first author was partially supported NSF Grant DMS9650266 and the second author by NSF Grant DMS9626330.

2. Construction of the manifold

The proof of Theorem 1 is constructive; we will find a smooth manifold homeomorphic to ∗CP^{2}# ∗ RP^{4}. The construction uses a homology sphere satisfying the conclusion of the following lemma, whose proof will be given in the next section.

Lemma 2.1. There is a homology 3–sphere Σ^{3} with the following properties.

(i) Σ is obtained by ±1 surgery on a knot K in S^{3}. (ii) The Rohlin invariant μ(Σ) = 1 (mod 2). (iii) Σ admits a free, orientation preserving involution τ, which is isotopic to the

identity.

Different Σ’s could in principle give rise to different smooth structures on ∗CP^{2}#∗ RP^{4}, but we know of no way to tell them apart. The situation is quite analogous to that for the fake RP^{4}’s constructed in [FS81].

Proof of Theorem 1. Let Σ be a homology 3–sphere as described in the lemma; choose an orientation on Σ so that it becomes surgery on a knot with coefficient = +1. Items (i) and (ii) are the ingredients in Freedman’s construction [Fre82] of ∗CP^{2}. That is, let Y be the result of adding a 2–handle to B^{4} along K, with framing 1, then ∂Y = Σ and

∗CP^{2} = Y ∪_{Σ} Δ^{4}

where Δ^{4} is a contractible 4-manifold with boundary −Σ. (The sign of the framing is not really important, for the difference between ∗CP^{2} and ∗\overline{CP}^{2} will disappear when we connect sum with ∗RP^{4}.) The non-trivial μ-invariant is readily identified with the Kirby-Siebenmann invariant of ∗CP^{2}.

Now items (ii) and (iii) are exactly the ingredients for the construction of ∗RP^{4} given in [Rub84], i.e.

∗RP^{4} = Δ^{4}/(x ∈ Σ ∼ τ(x)) = (Σ/τ ˜× I) ∪_{Σ} Δ^{4}

(The authors of [HKT94] seem to have been unaware of this earlier construction of ∗RP^{4}; compare the discussion in [Kir97, Problem 4.74].)

Let X be the smooth manifold obtained as the union of Y and the mapping cylinder of the orbit map of the free involution τ on Σ, i.e.

X = Y ∪_{Σ} (Σ/τ ˜× I) = Y/(x ∈ Σ ∼ τ(x)).

Then X is manifestly smooth, and we claim that it is homeomorphic to ∗CP^{2}#∗RP^{4}. This seems quite plausible, for the construction amounts to performing a sort of connected sum, where instead of removing disks and gluing, we remove the ‘pseudodisc’ Δ^{4} and glue up. Unfortunately, we do not know an elementary proof, and must appeal to the homeomorphism classification theorem of [HKT94].

According to that work, the manifold ∗CP^{2}# ∗ RP^{4} is distinguished among nonorientable manifolds with π_{1} = Z_{2} by having b_{2} = 1, trivial Kirby-Siebenmann invariant, and by a codimension-2 Pin^{c} Arf-invariant. (The other possible manifolds, up to homeomorphism, with the same homology are CP^{2}#RP^{4}, ∗CP^{2}#RP^{4}, and CP^{2}# ∗ RP^{4}.) The Arf-invariant, whose value for ∗CP^{2}# ∗ RP^{4} is ±3 (mod 8), is that of a surface pulled back from CP^{N} via a map φ : X → CP^{N+1} which classifies c_{Φ} of the (primitive) Pin^{c} structure Φ.

A (topological) Spin^{c} structure on ∗CP^{2} also determines such a map, say φ′; it is easy to see that (in terms of the decomposition of ∗CP^{2} given above) that φ′ can be taken to be smooth on Y, and constant on Δ^{4}. To be more concrete, the dual surface F could be taken as a Seifert surface of K, capped off in the 2-handle. The Arf invariant of F (in ∗CP^{2}) is 4 (mod 8), as can be seen from this description of F, or by using Rohlin’s theorem as in [HKT94].

The Pin^{c} structure on ∗RP^{4} has for its characteristic class the non-trivial class in H^{2}(∗RP^{4}; Z). This class is ‘dual’ to a surface in ∗RP^{4} which again may be assumed to lie in ∗RP^{4} − Δ^{4}. By the homotopy invariance of the Arf-invariant for Pin^{−} structures, Arf(∗RP^{4}) ≡ Arf(RP^{4}) ≡ ±1 (mod 8). There is a unique Pin^{c} structure on Σ, so the Pin^{c} structures on ∗RP^{4} − Δ^{4} and ∗CP^{2} − Δ^{4} glue up to give a Pin^{c} structure Φ_{X} on X. The characteristic class c_{Φ_{X}} is clearly dual to the disjoint union of surfaces lying in the two pieces of X, so the Arf invariant is ±4±1 ≡ ±3 (mod 8), just as for ∗CP^{2}# ∗ RP^{4}. Since X is smooth, its Kirby-Siebenmann invariant is trivial, and so X is homeomorphic to ∗CP^{2}# ∗ RP^{4}.

The additional remark about the universal cover of X being standard may be seen as follows(cf. [FS81]). By the construction of X, its cover X̃ ≅ Y ∪_{τ} \overline{Y} ≅ Y ∪ \overline{Y} since τ is isotopic to the identity. On the other hand, Y ∪ \overline{Y} is obtained by adding two 2-handles to B^{4}, together with a 4-handle. The first is added along K, with framing 1, and the second is added along a meridian of K, with framing 0. (This is a standard argument in handle theory, see for example [Kir89].) It is then easy to unknot K, by repeatedly sliding over the 0-framed handle, resulting in a standard picture of CP^{2}#\overline{CP}^{2}.

3. Proof of Lemma 2.1

In this section, we give two examples of homology spheres satisfying the conclusions of Lemma 2.1. Both examples are Brieskorn spheres, i.e. Seifert-fibered homology spheres of the form Σ(p, q, r), where p, q, and r are relatively prime odd numbers. The involution τ is nothing more than multiplication by −1 ∈ S^{1} in the natural circle action on Σ(p, q, r). The condition that the numbers p, q, and r be odd guarantees that τ is free; since −1 is contained in a circle, the involution is isotopic to the identity.

There are many Brieskorn spheres which are integral surgery on a knot–for some examples see [KT90, MM97] or adapt the technique of [CH81]. For most of these constructions one of the indices turns out to be even. One construction is given below, where it is shown that adding a handle (along the curve denoted γ) to the Brieskorn sphere Σ(5, 9, 13) yields S^{3}. Turning the picture upside down shows that Σ(5, 9, 13) is integral surgery on a knot in S^{3}. As remarked in the proof of Theorem 1, it doesn’t matter whether the coefficient is positive or negative. Again, the μ-invariant is 1 (from the picture just after blowing down the first −1 curve), so this example proves the lemma.

Another construction from the literature which provides Seifert fibered spaces is rs(p+q)^{2} +pq surgery on the knot denoted K_{p,q}(r, s) in the recent paper [MM97, §9].

Choosing p = −13, q = 23, r = 3, and s = 1 gives the homology sphere Σ(3, 13, 23) as +1 surgery on a hyperbolic knot. Since μ(Σ(3, 13, 23)) = 1, this manifold gives an example which yields the proof of Lemma 2.1. This is the only example of a μ-invariant 1 homology sphere constructible by this method found by a moderately long computer search. It is possible to give a Kirby-calculus proof that Σ(3, 13, 23) is surgery on a knot similar to the one for Σ(5, 9, 13); aficionados of the subject may wish to check if the knot is the same as the one in the knot from the paper [MM97].

References

[CH81] A. Casson and J. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981), 23–36.

[Fre82] M. Freedman, The topology of four–dimensional manifolds, J. Diff. Geo. 17 (1982), 357– 432.

[FS81] R. Fintushel and R. Stern, An exotic free involution on S^{4}, Annals of Math. 113 (1981), 357–365.

[HKT94] I. Hambleton, M. Kreck, and P. Teichner, Nonorientable 4-manifolds with fundamental group of order 2., Trans. A.M.S. 344 (1994), 649–665.

[Kir89] R.C. Kirby, Topology of 4-manifolds, Lecture Notes in Math., vol. 1374, Springer-Verlag, 1989.

[Kir97] R.C. Kirby, Problems in low–dimensional topology, Geometric Topology (W. Kazez, ed.), American Math. Soc./International Press, Providence, 1997.

[KT90] John Kalliongis and Chichen M. Tsau, Seifert fibered surgery manifolds of composite knots, Proc. Amer. Math. Soc. 108 (1990), 1047–1053.

[MM97] Katura Miyazaki and Kimihiko Motegi, Seifert fibred manifolds and Dehn surgery, Topology 36 (1997), 579–603.

[Rub84] D. Ruberman, Equivariant knots of free involutions of S^{4}, Top. Appl. 18 (1984), 217–224.

[Teich96] P. Teichner, On the star-construction for topological 4-manifolds, Geometric Topology (W. Kazez, ed.), American Math. Soc./International Press, Providence, 1997, 300–312.

Daniel Ruberman, Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254

  E-mail address: ruberman@binah.cc.brandeis.edu

Ronald J. Stern, Department of Mathematics, University of California, Irvine, California 92697

  E-mail address: rstern@math.uci.edu