CONNECTED SUM OF COMPACT

ANALYTIC HYPERSURFACES

Fabrizio Broglia^{∗} Domenico Luminati

Dipartimento di Matematica, Università di Pisa Dipartimento di Matematica, Università di Trento

March, 28, 1996

Abstract. A construction is given to find a global analytic [resp. algebraic] equation for the connected sum of two compact, analytic [resp. algebraic] hypersurfaces having global equation. As an application we give explicit algebraic equations for a model of the connected sum of k copies of ℝℙ^{3} in ℝ^{5}.

Introduction

Methods to obtain algebraic models of the connected sum of manifolds are well known (see [3]), but the dimension of the euclidean space where such models exist is usually very high. In the first section of this paper we give a construction that, in the case of compact hypersurfaces, produces a model in the same euclidean space. Moreover, the degree of the model is exactly the sum of the degrees of the two given hypersurfaces.

Since the proofs involve only the fact that the given hypersurfaces are analytic, the statements are all given in the C^{ω} case.

In the second section we use such a construction to find explicit algebraic models of the connected sum of k copies of ℝℙ^{3}’s in ℝ^{5}. Clearly such models do exist, in fact any compact, orientable 3-manifold can be C^{∞}-embedded in ℝ^{5} [6] hence, by Theorem 14.1.4 of [2], it can be isotopically approximated by a smooth algebraic variety in ℝ^{5}. This means, in particular, that any compact, orientable 3-manifold has an algebraic model in ℝ^{5}. In the case of the connected sum #_{k}ℝℙ^{3} it is easily seen by an argument due to Whitney [10] that 5 is the smallest embedding dimension.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 14P25.

Key words and phrases. anlytic manifold, algebraic model, connected sum, projective space.

We wish to thank the referee for his suggestions in improving the redaction of the paper.

The authors are partially supported by Eurocontract ERBCHRXCT940506 and by M.U.R.S.T.

∗Member of G.N.S.A.G.A. of C.N.R.

1. Connected sum of hypersurfaces

Let M be be an analytic set and let N ⊂ M − ∂M be a hypersurface We will say that N is a regular hypersurface if there exists a C^{ω} function f : M → ℝ such that 0 is a regular value for f and N = f^{−1}(0) (in particular N ⊂ M −Sing M). Such an f will be called an equation for N. Clearly the expression “regular hypersurface” is adopted only for convenience and to remind that one is speaking about the inverse image of a regular value of a function.

Let N_{1} and N_{2} ⊂ M be regular hypersurfaces in M and let f_{1}, f_{2} be proper equations for N_{1} and N_{2} respectively. Suppose that N_{1} ∩ N_{2} is constituted by a single point p.

Lemma 1.1. Let N′_{2} be the connected component of N_{2} containing p and call A_{+} = {f_{1} > 0} and A_{−} = {f_{1} < 0}. In the above situation N_{1} and N_{2} are tangent at p and N′_{2} − {p} is contained either in A_{+} or A_{−}.

Proof: N′_{2} − {p} is connected (in the case dim N_{2} = 1 use the fact that N_{2} is compact hence N′_{2} ≅ S^{1}) so it must be contained in one of A_{+} or A_{−}. From this follows that N_{1} and N_{2} are tangent at p. In fact if it were not then N_{2}(p) − {p} should have points in both A_{+} and A_{−}. □

So, without loss in generality, we can suppose df_{1}(p) = −df_{2}(p). For all ε ∈ ℝ let N_{ε} denote the set defined by the equation g_{ε} = f_{1}f_{2} − ε = 0. We will prove the following:

Proposition 1.2. There exists δ > 0 such that for all 0 < ε < δ N_{ε} is a regular hypersurface diffeomorphic to a connected sum N_{1}#N_{2} and g_{ε} is a proper, regular equation for it.

Remark. Suppose that N_{1} and N_{2} are smooth complete intersections of codimension k, and suppose that they are defined respectively by f_{1}, g_{2}, …, g_{k} and f_{2}, g_{2}, …, g_{k} (i.e. k −1 equations are the same for both the N_{i}), being f_{i} proper. Suppose finally that they intersect at a single point p and that df_{i}(p) have opposite directions. Then for ε > 0 small enough the set defined by f_{1}f_{2} − ε, g_{2}, …, g_{k} is diffeomorphic to a connected sum N_{1}#N_{2}.

To see this, it is enough to apply Proposition 1.2 considering the analytic set M′ defined by g_{2}, …, g_{k} as the ambient space (see Fig. 1).

\begin{matrix}^{f_{1} = 0 f_{2} = 0} & f_{1}f_{2} = ε \\ M′ \end{matrix} ; M′ Fig. 1

Proof of Proposition 1.2. First we prove that for ε ≠ 0 small enough, N_{ε} is regular and f_{1}f_{2} − ε is a regular equation for it. To see this, note that f_{1}f_{2} has no critical points on N_{0} = N_{1} ∪N_{2} other then p, hence for all neighborhood U of p there exists a neighborhood V of N_{0} − U, that contains no critical points for f_{1}f_{2}. Since f_{1} and f_{2} are proper functions, their product f_{1}f_{2} is proper too, hence N_{ε} ⊂ U ∪ V for ε small enough. So it is enough to show that we can choose U containing no critical points other then p, that is p is an isolated critical point for f_{1}f_{2}. But this follows by an easy application of the curve selection lemma (see [7], [1] for the lemma in the algebraic case or [8] for the general case).

Since df_{1}(p) = −df_{2}(p) ≠ 0 and since, by Lemma 1.1, N_{1} and N_{2} stand on opposite sides at p, we can find a neighborhood U of p and analytic coordinates x = (x_{1}, …, x_{n}), y on U, centered at p and such that f_{1}(x, y) = h(x) + y and f_{2}(x, y) = q(x, y)(h(x) − y), where h is a non negative analytic function vanishing only at 0, and q is a positive analytic function. Hence in such coordinates f_{1}f_{2}(x, y) = q(x, y)(h^{2}(x) − y^{2}).

Let c be a positive number and U′ = {(x, y) | |y| ≤ c}. U′ is an analytic manifold with boundary and the two functions g_{1}, g_{2} : U′ → ℝ defined by g_{1}(x, y) = h^{2}(x)−y^{2} and g_{2}(x, y) = f_{1}f_{2}(x, y) = q(x, y)(h^{2}(x) − y^{2}) have the same sign and the same zero set. Is a well known fact (see for instance [1] for the algebraic case and [8] for the general case) that in such hypothesis there exists δ > 0 such that for all 0 < ε ≤ δ the sets

V_{1} = {(x, y) | g_{1}(x, y) ≤ ε}, V_{2} = {(x, y) | g_{2}(x, y) ≤ ε}

are diffeomorphic.

The same argument shows that, for ε and c small enough, the set {x | h^{2}(x) ≤ c^{2} + ε} is diffeomorphic to the disk D^{n}. Since the functions y and its restriction to the boundary of V_{2} have no critical points on V_{2}, then V_{2} is a cylinder, that is to say, V_{1} and V_{2} are both diffeomorphic to D^{n} × [−c, c].

Finally, observe that ∂f_{1}f_{2}/∂x is not zero at all points of N_{0} other then p, hence, for ε small enough, the manifold N_{ε} is transversal to the the boundary of U′. This implies that N_{ε} − Ů′ is diffeomorphic to N_{0} − Ů′. Moreover, for c small enough, the two sets N_{i} ∩ U′ are both diffeomorphic to the set {x | h(x) ≤ c^{2} }, which is diffeomorphic to the disk D^{n}.

This is enough to say that N_{ε} is diffeomorphic to the manifold obtained gluing N_{1} − Ů′ and N_{2} − Ů′ along their boundaries with a diffeomorphism, which extends to a diffeomorphism of the two disks N_{i} ∩ U′, that is to say N_{ε} is diffeomorphic to a connected sum N_{1}#N_{2}. □

Remark. In the case that N_{1} and N_{2} are both orientable, observe that if they are oriented in such a way that the gluing diffeomorphism is orientation reversing, then N_{ε} is just the oriented connected sum. So if M − Sing M is oriented, and N_{1} and N_{2} are oriented according to their equations (i.e. the gradient of the equation, with respect to some Riemannian metric on M − Sing M, and an oriented bases of the tangent space to the hypersurface are an oriented bases of the tangent space to M − Sing M at each point of the hypersurface) then the resulting hypersurface N_{ε}, equipped with the orientation induced by the equation g_{ε}, is diffeomorphic to the oriented connected sum N_{1}#N_{2}.

Remark. In the case of hypersurfaces N_{1}, N_{2} of Euclidean space fulfilling the hypothesis of Proposition 1.2, the construction gives a model of N_{1}#N_{2} embedded in the same Euclidean space. Furthermore, in the case of algebraic hypersurfaces of degree d_{1} and d_{2} the construction gives a model for N_{1}#N_{2} of degree d_{1} + d_{2}.

Remark. Finally observe that whenever two compact hypersurfaces N_{1}, N_{2} ⊂ ℝ^{n+1} are given, one can find new ones N′_{1}, N′_{2} such that their intersection is a single point and N′_{i} is diffeomorphic to N_{i}. This fact can be seen as follows. Chose a unit vector v such that the function h_{v} : x ↦ ⟨v, x⟩ has an isolated maximum point p_{1} on N_{1} (in fact, since N_{1} is compact, there exists an open and dense subset U ⊆ S^{n} such that h_{v} is a Morse function for N_{1}, see for instance [5]). Let p_{2} a minimum point

for h_{v}_{N_{2}} and let d_{1} = min{∥x_{1} − p∥ | x ∈ N_{1}, h_{v}(x) = h_{v}(p_{1}), x ≠ p_{1}} and

d_{2} = max{∥x − p_{2}∥ | x ∈ N_{2}}. If τ is the translation x ↦ x + p_{2} − p_{1} and λ is the dilatation x ↦ p_{2} + (x − p_{2})d_{1}/2d_{2}, then it is easy to check that N′_{1} = τ(N_{1}) and N′_{2} = λ(N_{2}) are in the desired position (N′_{1} ∩ N′_{2} = {p_{2}}). Observe that, in the algebraic case, this procedure does not change the degree of the equations.

2. Connected sum of ℝℙ^{3}’s

As announced in the introduction, we apply the construction of §1 to find an explicit algebraic model of the connected sum of ℝℙ^{3}’s in ℝ^{5}. As already observed, such a model does exist and ℝ^{5} is in fact the smallest euclidean space where it can be found. This follows by a classical result due to Whitney [10] saying that all sections of the normal bundle to an embedding of ℝℙ^{2} in ℝ^{4} has a zero, hence:

Proposition 2.1. Every 3-manifolds containing an embedded projective plane cannot be embedded into ℝ^{4}.

Proof: If such an embedding existed, its normal bundle should be trivial, giving a never zero section of the normal bundle of the ℝℙ^{2} contained in the 3-manifold. □

In particular the connected sum of ℝℙ^{3}’s cannot be embedded in ℝ^{4}.

To construct an algebraic model of ℝℙ^{3}#ℝℙ^{3} in ℝ^{5}, we start from a well known model of ℝℙ^{3} ⊂ ℝ^{5}. ℝℙ^{3} is diffeomorphic to the unitary tangent bundle of the sphere S^{2} (see for instance [9]), hence it has obvious equations in ℝ^{6} = ℝ_{x}^{3} × ℝ_{y}^{3}:

(2-1)  ∥x∥^{2} = 1,  ∥y∥^{2} = 1,  ⟨x, y⟩ = 0.

This means that ℝℙ^{3} is an algebraic subset of S^{2} × S^{2}, but S^{2} × S^{2} is algebraically embedded in ℝ^{5} (as the hypersurface obtained by rotating a sphere S^{2} around a 2-plane), hence the same holds for ℝℙ^{3}. By an easy calculation one can write down the following equations for ℝℙ^{3} ⊂ ℝ^{5}:

(∥x∥^{2} − (R^{2} + r^{2}))^{2} + 4R^{2}(x_{4}^{2} + x_{5}^{2}) = 4R^{2}r^{2}, (2-2)

x_{1}(∥x∥^{2} − (R^{2} + r^{2})) + 2Rx_{2}x_{4} + 2Rx_{3}x_{5} = 0. (2-3)

The former is the equation of the torus S^{2} × S^{2}, spanned by the rotation of the 2-sphere of radius r, centered at (R, 0, 0, 0, 0) and contained in the 3-space of coordinates (x_{1}, x_{4}, x_{5}) around the 2-space of coordinates (x_{4}, x_{5}), the latter is derived from the third of (2-1). More precisely, consider the mapping Φ : ℝ^{6} → ℝ^{5}

Φ(x, y) = (x_{1}(R + ry_{1}), x_{2}(R + ry_{1}), x_{3}(R + ry_{1}), ry_{2}, ry_{3})

then Φ _{S^{2}×S^{2}} is a diffeomorphism with the hypersurface defined by (2-2), and the second equation is the push forward via Φ of the third of the (2-1).

Call T ⊂ ℝ^{5} the algebraic variety defined by (2-2), M the one defined by (2-3) and N = T ∩ M. Is not hard to see that the gradients of the two equations are independent at each point of N, that is N is a smooth subvariety of ℝ^{5} which is a complete intersection.

Remark. It is also easy to prove that the mapping Φ is a biregular isomorphism between ℝℙ^{3} ⊂ ℝ^{6} and N, and since the embedding of ℝℙ^{3} in ℝ^{6} is biregular, then N is a biregular embedding of ℝℙ^{3} in ℝ^{5}.

The idea to construct equations for ℝℙ^{3}#ℝℙ^{3} is quite simple. Let p ∈ N, suppose to be able to construct another torus T_{1} such that the following hold:

(A) T ∩ T_{1} = {p}, that is the tori are tangent at p; (B) N_{1} = T_{1} ∩ M is diffeomorphic to ℝℙ^{3}.

Let f be the function defining T and f_{1} the one defining T_{1}, then, according to Proposition 1.2 and the subsequent remark, for ε small enough the equation

(2-4)  ff_{1} − ε = 0

will define in M a manifold diffeomorphic to N#N_{1}, that is (2-4) and (2-3) are algebraic equations for ℝℙ^{3}#ℝℙ^{3} ⊂ ℝ^{5}.

The problem is so reduced to find a torus T_{1} fulfilling conditions (A) and (B). In order to do this we prove that we can approximate the torus T by a family of tori T_{ε} that are all tangent to T at a fixed point p ∈ T; by transversality arguments, it will follow that, for ε small enough, the tori belonging to such a family definitively satisfy the desired conditions.

Proposition 2.2. Let p ∈ T, then there exists a family {T_{ε}}_{ε>0} of tori such that: (1) T_{ε} ∩ T = {p} (i.e. T_{ε} is tangent to T at p);

(2) for all neighborhood U of T there exist δ > 0 such that T_{ε} ⊂ U for all 0 < ε < δ.

p p Fig. 2

Proof: Let c^{+} = (R, 0, 0, 0, 0), c^{−} = (−R, 0, 0, 0, 0) and let S^{+}, S^{−} be the 2-spheres of radius r, centered at c^{+}, c^{−} respectively and contained in the 3-space E, of coordinates (x_{1}, x_{4}, x_{5}). It is clear that S^{+}, S^{−} ⊂ T, and T is spanned by the rotation of either S^{+} or S^{−} around the 2-space A of coordinates (x_{4}, x_{5}) (i.e. T is the orbit of either S^{+} or S^{−} under the action of the group of rotations of ℝ^{5} leaving A fixed). Without loss in generality we can suppose p ∈ S^{+}. For all ε ∈ (0, r) let S_{ε}^{+} be the 2-sphere in E of radius r − ε and centered at c^{+} + ε(p − c^{+})/r (i.e. S_{ε}^{+} is inside S^{+} and tangent at S^{+} in p) and S_{ε}^{−} the 2-sphere of radius r − ε centered at c^{−}. Finally let B_{ε} ⊂ E be the 2-plane of symmetry of S_{ε}^{+}, S_{ε}^{−} and let T_{ε} be the torus spanned by the rotation of either S_{ε}^{+} or S_{ε}^{−} around B_{ε}. We claim that the family {T_{ε}} has the required properties (see Fig. 2).

Clearly, since S_{ε}^{+} → S^{+} and B_{ε} → A as ε → 0, (2) follows by compactness arguments.

For what concern (1), we remark that T_{ε} is a union of S^{2} spanned by the rotation of either a point p^{+} ∈ S_{ε}^{+} or p^{−} ∈ S_{ε}^{−} around their axis B_{ε}, hence the thesis is a consequence of the following lemma:

Lemma 2.3. Let p^{+} ∈ E be a point inside S^{+}, p^{−} ∈ E a point inside S^{−} and let B ⊂ E be the axis of the segment joining p^{+} and p^{−}. Let S ⊂ ℝ^{5} be the 2-sphere spanned by the rotation around B of either p^{+} or p^{−}. Then S is contained inside T. Moreover, if p^{+} ∈ S^{+} then S is tangent to T at p^{+}.

Proof: Suppose that p^{+} is inside S^{+}. Let X ∈ S, then X = σ(p^{+}), being σ a rotation of ℝ^{5} leaving B fixed. Let τ be a rotation such that τ(X) ∈ E ∩ {x_{1} ≥ 0} and leaving A fixed (clearly such a τ does exist). Let us call Y = τ(X). It is obvious that to prove the lemma is equivalent to show that Y is inside S^{+}.

Let x_{i}(X) denote the i^{th} coordinate of X, then (x_{1}(X), x_{4}(X), x_{5}(X)) ∈ E belongs to the segment joining p^{+} to p^{−} and, since the rotation τ lives A fixed, (x_{4}(Y), x_{5}(Y)) = (x_{4}(X), x_{5}(X)) hence Y belongs to the cylinder C parallel to the x_{1} axis and based at the segment joining (0, x_{4}(p^{+}), x_{5}(p^{+})) to (0, x_{4}(p^{−}), x_{5}(p^{−})).

\begin{aligned}x_{5} & & ^{x_{4} x_{1}} \\ & p^{+} \end{aligned} ; p^{−} ; q^{−} ; A ; B ; C Fig. 3

Let q^{−} denote the symmetric of p^{−} in E with respect to A. Suppose that A ∩ B is a line, since Y = τσ(p) and τσ leaves A ∩ B fixed then Y belongs to the circular cylinder in E having A∩B as axis and passing through p^{+}. Together with what we proved above, this means that Y belongs to the arc of the circle passing through p^{+}, p^{−} and q^{−}, and joining p^{+} to q^{−}, from which the thesis follows.

If A and B are parallel or equal, it is easily seen that the point Y belongs to the segment joining p^{+} to q^{−}. Once again the thesis follows.

Finally, if p^{+} ∈ S^{+}, observe that X ∈ T if and only if Y ∈ S^{+}. But the arc containing Y touches S^{+} exactly at p^{+}, and Y = τσ(p^{+}) = p^{+} implies that σ is the identity, hence X = p^{+}. □

This ends the proof of Proposition 2.2. □

Remark. Lemma 2.3 can be obviously restated for the embedding S^{p}×S^{q} ⊂ ℝ^{p+q+1} obtained by the rotation of an S^{q} around a q-plane, and the same kind of arguments, used in the above proof, proves this general case too.

Clearly the torus T_{ε} constructed in Proposition 2.2 is obtained from T changing the parameters R and r and acting on it by an isometry of ℝ^{5}, hence, if f(R, r, x) denotes the function defining the torus T, that is the left hand side of equation (2-2), then T_{ε} has an equation of the form f(R_{ε}, r_{ε}, φ_{ε}(x)) = 0, where

R_{ε} = \frac{1}{2} ∥c^{+} − c^{−} + \frac{ε}{2}(p − c^{+})∥ , r_{ε} = r − ε,

and φ_{ε} is an isometry of ℝ^{5}, which is a C^{∞} function of ε. We now prove the following proposition:

Proposition 2.4. If ε is sufficiently small, then T_{ε} ∩ M is diffeomorphic to ℝℙ^{3}.

Proof: Let M̃ denote M − U, being U a small ball around 0 such that T is far from ∂(M − U). Consider the mapping F : M̃ × ℝ → ℝ × ℝ defined by:

F (x, ξ) = (f(R_{ξ}, r_{ξ}, φ_{ξ}(x)), ξ).

F is of class C^{∞}, for ξ sufficiently small, and proper. Furthermore F^{−1}(0, ε) = (T_{ε} ∩ M) × {ε}. Since T_{0} = T and T ∩ M is a complete intersection, (0, 0) is a regular value for F. By Ehresmann’s fibration theorem (see for instance [4] or [5]), if ε is sufficiently small, F^{−1}(0, ε) is diffeomorphic to F^{−1}(0, 0) = (T_{0} ∩ M) × {0}. Since T_{0} = T is diffeomorphic to ℝℙ^{3}, the thesis follows. □

Proposition 2.2 and Proposition 2.4 ensures that one can find a torus T_{1} such that (A) and (B) hold, hence one can find algebraic equations for ℝℙ^{3}#ℝℙ^{3} in ℝ^{5}.

Remark. Let f and f_{1} be the equations of T and T_{1} respectively. By the way we constructed T_{1}, the equation f_{1} has the same degree as f, that is 4. Hence the two equations we found, that is (2-3) and (2-4) have degree respectively 3 and 8, in other words we found ℝℙ^{3}#ℝℙ^{3} as a smooth algebraic subset of ℝ^{5} of degree 24.

It is clear from the proof in §1, Proposition 1.2 can be generalized as follows:

Proposition 2.5. Let N_{1}, …, N_{k} ⊂ M be regular hypersurfaces, and f_{1}, …, f_{k} proper, regular equations for them. Suppose that

N_{i} ∩ N_{j} = \{ \begin{matrix}\{p_{i}\} & if |i − j| = 1 \\ ∅ & otherwise \end{matrix} and df_{i}(p_{i}) = −df_{i+1}(p_{i}) ∀ i = 1, …, k − 1.

Then there exists δ > 0 such that for all 0 < ε < δ the set N_{ε} = {x ∈ M | g_{ε}(x) = 0}, where g_{ε} = f_{1} ⋯ f_{k} −ε, is a regular hypersurface diffeomorphic to the connected sum N_{1}# ⋯ #N_{k} and g_{ε} is a proper, regular equation for it.

Remark. Using this proposition one can iterate the construction of the torus T_{1}, to find algebraic equations for the connected sum #_{k}ℝℙ^{3} in ℝ^{5}. More precisely let p_{1} ∈ N and T_{1} be as above, that is T_{1}∩M is diffeomorphic to ℝℙ^{3}, and T_{1} is tangent to T at p_{1}. Now, let p_{2} ∈ T_{1} ∩M, p_{2} ≠ p_{1} the same arguments used to construct T_{1} can be used to construct a torus T_{2} such that T_{2}∩M is diffeomorphic to ℝℙ^{3} and T_{2} is tangent to T_{1} at p_{2}. This process can be iterated k times. Observe that each time a new torus is constructed, it is inside the previous one (except for the tangency point), hence T_{i} ∩ T_{j} = ∅ if |i − j| ≥ 2. By Proposition 2.5, if f, f_{1}, …, f_{k−1} are the equations of these tori, than for a suitable choice of ε, ff_{1} ⋯ f_{k−1} − ε together with (2-3) are equations for the connected sum of k copies of ℝℙ^{3}’s. Once again all the equation f_{i} have the same degree as f, hence we find #_{k}ℝℙ^{3} in ℝ^{5} as a smooth algebraic surface of degree 12k, which is complete intersection.

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F. Broglia, Dipartimento di Matematica, Università di Pisa, v. F. Buonarroti n. 2, 56127 Pisa, Italy.

E-mail address: broglia@dm.unipi.it

D. Luminati, Dipartimento di Matematica, Università di Trento, v. Sommarive n. 14, 38050 Povo, Trento, Italy.

E-mail address: luminati@science.unitn.it