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Celebratio Mathematica

Hassler Whitney

Topology

The Whitney trick

by Rob Kirby

The Whit­ney trick is a meth­od for re­mov­ing points of in­ter­sec­tion between two sub­man­i­folds. It can be seen in its most ele­ment­ary form in Fig­ure 1, in which it is ob­vi­ous that the two points of in­ter­sec­tion can be re­moved by an iso­topy (a 1-para­met­er fam­ily of em­bed­dings) of the arc labeled \( P^p \) which pulls the arc across the disk \( D \). (Note that \( x \) and \( y \) have op­pos­ite signs.) More gen­er­ally the Whit­ney trick is used to re­move a pair of in­ter­sec­tions, \( x \) and \( y \), between two man­i­folds \( P^p \) and \( Q^q \) which are em­bed­ded in an am­bi­ent man­i­fold \( M^{p+q} \). To see how this is done, we first con­struct a mod­el, then show how to em­bed it in \( M \) (if pos­sible), and then sketch some ap­plic­a­tions of the Whit­ney trick.

Figure 1: The Whitney trick in the plane.

The mod­el is merely a sta­bil­iz­a­tion of the ex­ample in Fig­ure 1. We cross the plane in which \( D \) is em­bed­ded with \( \mathbb R^{(p-1) + (q-1)} \) so that the am­bi­ent space is just \( \mathbb R^{p+q} \), and then we cross the curve which in­cludes \( \alpha \) by \( \mathbb R^{p-1} \) to get an \( p \)-di­men­sion­al man­i­fold \( P \), and sim­il­arly cross with \( \mathbb R^{q-1} \) to get an \( q \)-man­i­fold \( Q \). These two man­i­folds still meet in two points \( x \) and \( y \), which are con­nec­ted in \( P \) by the ori­gin­al arc \( \alpha \) and in \( Q \) by the ori­gin­al arc \( \beta \). Note that the two arcs still bound a 2-di­men­sion­al disk \( D \), and that \( D \) lies in­side a lar­ger open disk \( \Delta \) in the plane. Also note that \( \Delta \) has a nor­mal \( (p-1)+(q-1) \)-plane bundle which splits as the dir­ect sum (also called “Whit­ney sum”) of a \( (p-1) \)-plane bundle which co­in­cides along \( \alpha \) with the nor­mal bundle of \( \alpha \) in \( P \), and an \( (q-1) \)-plane bundle which co­in­cides along \( \beta \) with the nor­mal bundle of \( \beta \) in \( Q \).

The plane iso­topy de­scribed in Fig­ure 1 eas­ily ex­tends to an iso­topy tak­ing place in the plane crossed with the \( p-1 \) co­ordin­ates of \( P \), as drawn for \( p=2 \) in Fig­ure 2 [e4]; noth­ing hap­pens with the oth­er \( q-1 \) co­ordin­ates.

Figure 2: The Whitney trick in dimension three.
© 2005 Scorpan.

Now this mod­el must be em­bed­ded in \( M^{p+q} \) so that the ac­tu­al man­i­folds \( P \) and \( Q \) and two points of in­ter­sec­tion \( x \) and \( y \) cor­res­pond to the man­i­folds and points in the mod­el.

If both \( P \) and \( Q \) are con­nec­ted, then the arcs \( \alpha \) and \( \beta \) ex­ist, and if \( P \) and \( Q \) are simply con­nec­ted (as they of­ten are in ap­plic­a­tions), then the arcs are unique up to ho­mo­topy. If \( M \) is simply con­nec­ted, then the disk \( D \) can be mapped in­to \( M \). If not, then \( x \) must be con­nec­ted by an arc (unique up to ho­mo­topy if \( P \) is simply con­nec­ted) to a base point \( x_0 \in P \) which is con­nec­ted by an arc to a base point \( z \in M \). Sim­il­arly with arcs to a base point \( y_0 \in Q \). It fol­lows that \( x \) then de­term­ines an ele­ment of \( \pi_1(M) \) by run­ning from \( z \) to \( x_0 \) to \( x \) to \( y_0 \) and back to \( z \). Now if \( x \) and \( y \) both rep­res­ent the same ele­ment of \( \pi_1(M) \), then we can still map a disk \( D \) in­to \( M \). (This is im­port­ant in prov­ing the \( s \)-cobor­d­ism the­or­em.)

Once \( D \) is mapped in­to \( M \), we can em­bed it if the di­men­sion of \( M \), \( p+q \), is five or more. Fur­ther­more, if each of \( p \) and \( q \) is three or more, then the em­bed­ding of \( D \) can be chosen to miss \( P \) and \( Q \) ex­cept along its bound­ary.

Now that \( D \) is em­bed­ded miss­ing \( P \) and \( Q \), it re­mains to find the em­bed­ding of the nor­mal bundle of \( D \). The nor­mal \( (p+q-2) \)-bundle to \( D \) (in fact, \( \Delta \)) in \( M \) can be split along \( \alpha \) as the nor­mal \( (p-1) \)-bundle to \( \alpha \) in \( P \) dir­ect sum the or­tho­gon­al \( (q-1) \)-bundle. That split­ting ex­tends across \( \Delta \). The only prob­lem re­main­ing is that this \( (p-1) \)-plane bundle may not co­in­cide with the \( (p-1) \)-plane bundle which is the nor­mal bundle to \( \beta \) in \( Q \).

The prob­lem re­duces to an arc of \( (p-1) \)-planes in \( \mathbb R^{(p-1) + (q-1)} \) which we want to iso­tope to the trivi­al arc, re­l­at­ive to the en­d­points. Note that the trivi­al arc, as in the mod­el, cor­res­ponds to \( x \) and \( y \) hav­ing op­pos­ite signs, so this is ne­ces­sary. Now, this is pos­sible be­cause the fun­da­ment­al group of the Stiefel man­i­fold of \( (p-1) \)-planes in \( \mathbb R^{p+q-2} \) is trivi­al when \( p > 2 \) (see [e3], p. 202). For more de­tails, see the ex­cel­lent de­scrip­tion in [e4].

Whit­ney de­veloped the Whit­ney trick in or­der to em­bed \( P^p \) in \( \mathbb R^{2p} \) [e1]. For \( p=2 \), this is easy. In high­er di­men­sions, \( P \) only im­merses in \( \mathbb R^{2p} \) (by gen­er­al po­s­i­tion), so for each double point, Whit­ney in­tro­duces in loc­al fash­ion an­oth­er double point of op­pos­ite sign (some thought is needed if \( P \) is non-ori­ent­able), and then uses the Whit­ney trick to re­move both points of in­ter­sec­tion.

A later, and cru­cial, use of the Whit­ney trick is in Smale’s proof of the \( h \)-cobor­d­ism the­or­em [e2].