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Celebratio Mathematica

Hassler Whitney

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 Pp 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 Pp and Qq which are em­bed­ded in an am­bi­ent man­i­fold Mp+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 R(p1)+(q1) so that the am­bi­ent space is just Rp+q, and then we cross the curve which in­cludes α by Rp1 to get an p-di­men­sion­al man­i­fold P, and sim­il­arly cross with Rq1 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 α and in Q by the ori­gin­al arc β. Note that the two arcs still bound a 2-di­men­sion­al disk D, and that D lies in­side a lar­ger open disk Δ in the plane. Also note that Δ has a nor­mal (p1)+(q1)-plane bundle which splits as the dir­ect sum (also called “Whit­ney sum”) of a (p1)-plane bundle which co­in­cides along α with the nor­mal bundle of α in P, and an (q1)-plane bundle which co­in­cides along β with the nor­mal bundle of β in Q.

The plane iso­topy de­scribed in Fig­ure 1 eas­ily ex­tends to an iso­topy tak­ing place in the plane crossed with the p1 co­ordin­ates of P, as drawn for p=2 in Fig­ure 2 [e4]; noth­ing hap­pens with the oth­er q1 co­ordin­ates.

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

Now this mod­el must be em­bed­ded in Mp+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 α and β 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 x0P which is con­nec­ted by an arc to a base point zM. Sim­il­arly with arcs to a base point y0Q. It fol­lows that x then de­term­ines an ele­ment of π1(M) by run­ning from z to x0 to x to y0 and back to z. Now if x and y both rep­res­ent the same ele­ment of π1(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+q2)-bundle to D (in fact, Δ) in M can be split along α as the nor­mal (p1)-bundle to α in P dir­ect sum the or­tho­gon­al (q1)-bundle. That split­ting ex­tends across Δ. The only prob­lem re­main­ing is that this (p1)-plane bundle may not co­in­cide with the (p1)-plane bundle which is the nor­mal bundle to β in Q.

The prob­lem re­duces to an arc of (p1)-planes in R(p1)+(q1) 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 (p1)-planes in Rp+q2 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 Pp in R2p [e1]. For p=2, this is easy. In high­er di­men­sions, P only im­merses in R2p (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].