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#### Topology of 4-Manifolds

**By Michael H. Freedman, Frank Quinn**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 1990 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-08577-7

All rights reserved.

ISBN: 978-0-691-08577-7

CHAPTER 1

*Basic tools*

In this chapter the principal methods for manipulation of immersions of surfaces are described. The raw material is provided by the immersion lemma 1.2, which is assumed as an axiom in Part I. This and the twisting operation of 1.3 are used to construct (immersed) Whitney disks in 1.4. These disks are used to define Whitney moves, which axe one of the principal geometric operations in topology of any dimension. Finger moves and connected sums are developed in sections 1.5 and 1.8 as ways to remove intersections, especially with Whitney disks. Intersection numbers, described in 1.7, give algebraic conditions under which the geometric operations can be used. One of the central ingredients of later constructions, transverse spheres, are introduced in 1.9. Finally section 1.10 describes "scorecards," a bookkeeping device useful for recording intersections among families of surfaces.

**1.1 Pictures in 4-space**

We picture 4-space as the familiar 3 space dimensions, and time. A picture of a 3-dimensional object therefore represents something in a slice ("the present") of **R4.** The strategy will be to compress as much as possible of the activity into this slice. Several conventions will be used in the pictures. Solid lines will represent things in the present time. Dotted lines represent things in the future, dashed lines are in the past. We suggest that the reader draw over the dotted lines in color (eg. blue = future, red = past). Color is much more effective in suggesting another dimension, but we cannot print the pictures in color. A line in the present which passes behind something is drawn thinner (since we cannot use the standard convention of drawing it dashed). Finally, we sometimes use a convention to avoid drawing things in the past or future. If a surface is drawn as a line, it is understood that it extends into the past and future simply as product with **R.**

In the first example a plane *A* is shown entirely in the present. *B* starts in the past, intersects the present in a line, and extends into the future. This is shown with and without the time extension convention.

Since *A* lies entirely in the present the intersection with *B* is easy to see: it is the single point of intersection with the line of *B* lying in the present.

**Exercise** Suppose c is an arc in a plane, and a neighborhood of the plane is parameterized as **R**2×{0} [subset] **R**2×**R**2. The *linking annulus* of the arc is the annulus *c*×*S1* [subset] **R**2 x **R**2. (1) Let c be the line of intersection of the plane B with the present, and draw the linking annulus. (2) Draw the *linking torus* of an intersection point; if a neighborhood of the point is parameterized as **R**2×**R**2, then the torus is S1 x S1. In the picture above this torus intersects the present in two circles, and extends to a copy of S1×D1 in both the past and the future.

**1.2 Framed immersons**

These provide models for images of surfaces mapped into 4-manifolds. Suppose *N* is a surface, and let *D, E* be disjoint copies of **R**2 in *N*. Form *N*×**R**2, and identify *D*×**R**2 with *E* x **R**2 by the isomorphism which switches the coordinates. This is a *plumbing* of *N* x **R**2 with itself. The identification space obtained by some finite number of disjoint plumbings is a smooth manifold. The image of *N* in this is embedded except at isolated double points, which have neighborhoods isomorphic to the picture drawn in 1.1.

A *framed immersion* of *N* into *M*4 is an isomorphism of an open set in *M* with such a plumbing. A framed immersion defines a map *N* -*M* with isolated double points, and to simplify notation we often refer to this map as the framed immersion. It is quite important, however, that a framed immersion specifies a neighborhood of the image as well as the image itself. When it is necessary to be precise we say that the map *extends* to a framed immersion. Note that the plumbing has boundary ([partial derivative]N)×**R**2, and the boundary of an open set in *M* is the intersection with [partial derivative]*M*. Therefore immersions under this definition are automatically "proper" in the sense that they take [partial derivative]*N* to [partial derivative]*M*.

Consider *D*2 as the unit disk in the complex numbers. If r: N - S1 is a map we can define an automorphism of *N*×*D2* by multiplying by *r(n)* in *{n}*×*D2*. Two framed immersions *differ by rotations* if one is obtained from the other by composing with such an automorphism of *N×D2*. Similarly we can let -1 [member of] *S0* act on **R**2 by [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]] so a map r: N - S0 = {[+ or -]1} defines an automorphism of *N*×**R2**. Two immersions *differ by sign* if they differ by an automorphism of this form.

**Immersion lemma.** Suppose f : N2 - M4 is a map.

(1) If N = D2 or **R** ×*I* and f extends to a framed immersion in a neighborhood of [partial derivative]N then there is a homotopy of f rel [partial derivative] N to a map which extends to a framed immersion, which near [partial derivative] N differs from the given one by rotations (N = D2) or sign (N = **R**×I).

(2) If N is the union of open sets U [union] V, and the restrictions of f extend to framed immersions which agree on U [intersection] V×**R**2, then f can be changed by isotopy on U rel U [intersection] V so that the result extends to a framed immersion of the union.

Note that any surface can be decomposed into disks and ribbons, so

(1) can be used to build up immersions of arbitrary surfaces. The signs and rotations in the normal direction define a normal vector bundle for the image.

The point of (2) is that *U* and *V* might individually be good, but the intersections between them not come from plumbings. When this *is* the case (ie. *f* defines an immersion—in the sense above—of *U* [union] *V*) then the pieces are then said to be "in general position," or "transverse."

This lemma provides the starting data for most of our constructions. In the smooth and PL categories it is one of the simplest cases of transversality and general position, and can be found in many textbooks. We will not give a proof here. The topological version is proved in section 9.4C, as a consequence of the smoothing theorem proved in chapter 8. It is a deep fact, and indeed depends on the whole development in the smooth or PL case.

The image of *N]×D2* in a plumbing of N×**R**2 is a regular neighborhood of the image of *N*. If each component of *N* has nonempty boundary then this regular neighborhood is also a regular neighborhood of a graph; choose a graph in *N* which has *N* as a regular neighborhood. At each intersection point add two embedded arcs, one in each sheet, joining the intersection point to the graph. Then the regular neighborhood of the image is isotopic to a regular neighborhood of this enlarged graph.

As a consequence of this description of the regular neighborhood we have a useful description of the fundamental group of the image: it is free, and if *N* is connected it is generated by loops passing through at most one intersection point. This is based on the assumption that components of *N* have nonempty boundary, but the π1 conclusion remains true if the closed components are spheres.

**1.3 Twisting**

The twisting operation changes an immersion so as to change the framing of the boundary. It can often be used to undo the rotations encountered in the immersion lemma.

Consider a standard *D2* [subset] *D4*. Cut out a square in the interior, and replace it by two pieces; a strip twisted about one edge, and a piece obtained by filling in straight lines between points on the opposite edge and corresponding points on the twisted strip.

As described, this object is completely in the present. Both the twisted strip and the fill-in piece have a line of intersection with the *D2*. Perturb the interior of the twisted strip into the past, and the interior of the fillin piece slightly into the future. The result is an immersion which has a single selfintersection point. Near the boundary it differs from the original by rotations, specifically two full twists; [+ or -]2 in π1S1 [equivalent] Z. This can be seen directly—one twist is evident in the twisted strip, the other comes from the shifting into the past and future—or it can be deduced from the presence of the single selfintersection point using the formula relating intersections and Euler numbers (1.7).

There is a technically more important "boundary twist" operation, but before considering that we give a corollary of interior twisting.

**1.3] A Corollary.***A map f: S2 - M4 is homotopic to a framed immersion if and only if ω2(f) = 0.*

Here the second Stiefel-Whitney class ω2 [member of] H2 (M; Z/2) is thought of as a homomorphism ω2: H2(M; Z/2) -Z/2), ω2(f) denotes the value of this on the image of the orientation class of *S2*.

**Proof:** Arrange, by homotopy, that *f* intersects a ball in *M* in a standard disk *D2 [subset] D4*. Let *M0* denote the manifold obtained by deleting the interior of the ball of radius ½, then *f* defines a map f0:D2 - M0 which is an immersion near the boundary. Apply the immersion lemma to find a homotopic immersion *f0'* which differs from the given one by rotations near the boundary. The number of full twists in the rotation function is equal mod 2 to ω.sub.2](f) so ω2(f) = 0 if and only there are an even number of full twists. In this case we can apply the twisting operation in the interior of the disk to obtain an immersion with nullhomotopic rotation function. A null-homotopy can be used to define a framing of the union of the standard disk and the twisted *f0',* giving a framed immersion.

To see the other direction (and the fact about ω2(f)) requires some background we will not review in detail. Briefly, when the rotations are nontrivial the standard disk and *f0'* fit together to define an "unframed" immersion of *S2.* The normal bundle of this immersion has the rotation function as clutching function, so the degree of the rotation function determines the bundle as an element of π2(**BO**(2)) [congruent to] **Z**. The degree mod two therefore determines the stable class in π2(**BO**(2)) [congruent to]**Z**/2. But this stable class depends only on the homotopy class of *f* (and the tangent bundle of *M*) and i s determined by ω2(f). Therefore if ω2(f) ≠ 0, f cannot be represented by an immersion with trivial normal bundle.

Now we turn to "boundary twisting." For this we need the notion of a surface *B* being immersed with part of its boundary on another surface *A.* The model for such a thing is built from two surfaces *A* and *B,* an embedded collection of arcs and circles *c [subset] [partial derivative]B*, and an embedding c×[-1,1] -*A.* Locally the model looks like:

Explicitly, choose a collar *c*×[0,1] [subset] *B,* then there are subsets *c* × [0,1] × [-1,1] [subset] *B* × [-1,1] and *c* × [-1,1] × [-1,1] [subset] *A* × [-1,1], We identify the first with the image in the second, by the map which switches the last two coordinates. This defines a 3-manifold neighborhood of the union *B* [union]*c**A,* and a 4-manifold is obtained by product with *I.* Immersions of this object are defined by open embeddings of plumbings disjoint from the juncture *c.*

Now consider the model for a disk embedded in *D*4 with part of its boundary on the standard 2-disk (ie. the local model pictured above). Cut out a square in *B,* with one edge on *A.* Replace this by a twisted strip, twisted about the edge on *A*, and a fill-in piece as above:

The result is a map *B'* which is a framed immersion of *D*2 - *c,* and near *A* can be parameterized as a standard immersion of a collar on *c* attached to *A.* The framings of these two immersions differ by rotation by one full twist near *c.* Note that *B'* has a new intersection point with *A.*

**1.3B Corollary.**Suppose c [subset] S1 is nonempty, c × [-1,1] - S is an embedding in a surface, A: S [union] c (S1 × [0,1)) - M4 is a framed immersion, and f: D2 - M is a nullhomotopy of the restriction of A to S1. Then there is a framed immersion of S [union]c D2 which agrees with A on S [union]c S1 and is homotopic rel boundary to f on D2.

**Proof:** Denote by M0 the complement of a small regular neighborhood of the image of *c.* The immersion *A* defines an immersion *A*0 of the complement of a neighborhood of *c* in *S,* disjoint union with S1 × [0,1), to *M*0. *f* extends the latter to a map of a disk. Applying the immersion lemma gives an immersion *B: D*2 -*M*0 homotopic to f which near *S*1 differs from the given immersion by rotations. Replace *c,* and apply the boundary twisting procedure near a point in *c* to cancel the rotations. The result is the desired framed immersion.

**Exercise** Suppose *T* is a surface, c [subset] [partial derivative]T and *c* × [-1,1] -*S* is an embedding. Let *A: S [union] c ([partial derivative]T* × [0,1)) -*M* be a framed immersion which extends to a map of *S [union]c T* to *M.* Give a criterion generalizing the two corollaries for the existence of a framed immersion of *S [union]c T* which agrees with *A* on *S [union]c [partial derivative]T* and is homotopic rel boundary to the given map on *T.* For this use the fact that a map of a surface can be approximated by an embedding on a neighborhood of an arc, with the image of the arc disjoint from all other surfaces. Recall also that surfaces can be reduced to disks by cutting along arcs and circles.

**1.4 Whitney disks**

Suppose immersed surfaces *A, B* intersect in two points, and there is an embedded 2-disk *W* with boundary on the union of the images, as shown:

This picture is the standard model for a neighborhood of a *Whitney disk.* The boundary of this disk (a union of two axes, one on each surface) is called a *Whitney circle* for the intersection points. Finally we say that a pair of intersection points *have opposite sign* if they have a Whitney circle; if there are embedded arcs joining them on *A* and *B* with a neighborhood isomorphic to a neighborhood in the model.

When *A, B,* and *M* have appropriate orientation data, and *A* and *B* are connected, then this definition of "opposite sign" agrees with the definition of signs for intersection numbers (see 1.7). If a half twist is inserted in *A* or *B* in the picture above then the intersection points have the "same sign," and the circle cannot appear as the boundary of a Whitney disk.

An application of Corollary 1.3B in the previous section yields:

**Lemma.** (Immersed Whitney disks) *Suppose c is a Whitney circle in A [union] B which is contractible in M. Then there is an immersed Whitney disk with boundary c.*

As above "immersed" means a map whose image has a neighborhood obtained by introducing plumbings into the standard model. *W* is allowed to intersect *A* and *B,* as well as itself.

*(Continues...)*

Excerpted fromTopology of 4-ManifoldsbyMichael H. Freedman, Frank Quinn. Copyright © 1990 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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