Celebratio Mathematica

Dmitry Bor­iso­vich served as a kind of To­po­logy Min­is­ter of the Arnold sem­in­ar,1 re­spons­ible for this area of math­em­at­ics, and ran the dis­cus­sion when to­po­lo­gic­al ques­tions arose — which happened fre­quently be­cause of the ubi­quity of To­po­logy in Al­geb­ra­ic Geo­metry, Mech­an­ics, and Dy­nam­ic­al Sys­tems.

In these cases, he usu­ally cla­ri­fied the situ­ation in a few words, and either com­pletely closed the ques­tion (if it was com­par­at­ively easy) or char­ac­ter­ized its genre and area, the pre­sumed com­plex­ity, and meth­ods by which it prob­ably should be at­tacked. The num­ber of prob­lems solved by sem­in­ar par­ti­cipants based on these hints was ap­par­ently very large; however Fuchs him­self nev­er ap­peared as a coau­thor — prob­ably, con­sid­er­ing this work too easy for him­self.

Only in two or three cases, when he did al­most all of the com­pound work, Fuchs wrote the cor­res­pond­ing art­icles (and, I be­lieve, it was Arnold who in­sisted that he do it with­in a fi­nite time).

In one of these works [2] he has cal­cu­lated the mod 2 co­homo­logy ring of the space of un­ordered con­fig­ur­a­tions of \( d \) dis­tinct points in the plane (and also re­lated struc­tures: Steen­rod al­gebra ac­tion, sta­bil­iz­a­tion as \( d \to \infty \), the Hopf al­gebra struc­ture on the stable ho­mo­logy and co­homo­logy, geo­met­ric real­iz­a­tion of gen­er­at­ors, etc). Arnold came to the con­sid­er­a­tion of this ring (or, rather, of the sim­il­ar co­homo­logy group with ar­bit­rary coef­fi­cients) from his study of su­per­pos­i­tions of al­geb­ra­ic func­tions; he made some first cal­cu­la­tions, but asked Fuchs to con­tin­ue them to the case of ar­bit­rary val­ues of \( d \) and di­men­sions of co­homo­logy groups. Based on Fuchs’ res­ults, Arnold proved some the­or­ems on the ab­sence of de­com­pos­i­tions of al­geb­ra­ic func­tions in­to su­per­pos­i­tions of easi­er func­tions. Since then this work (and its vari­ous gen­er­al­iz­a­tions) has seen nu­mer­ous oth­er ap­plic­a­tions: in geo­met­ric com­bin­at­or­ics, in­ter­pol­a­tion the­ory, com­plex­ity the­ory, etc.

The to­po­lo­gic­al study of con­fig­ur­a­tion spaces has de­veloped since then in­to a huge and very in­stru­ment­al area of to­po­logy, and the main meth­ods and keywords first used by Fuchs in this pi­on­eer­ing work are now stand­ard for these stud­ies.

A closely re­lated work is the cal­cu­la­tion of co­homo­logy rings of ordered con­fig­ur­a­tion spaces of com­plex num­bers, also ini­ti­ated by Arnold. Ac­cord­ingly to Arnold’s re­min­is­cences, the first work of this the­ory [e2] also was writ­ten with huge help from Fuchs, al­though in this case he does not ap­pear as an au­thor.

The oth­er work [1] is a cla­ri­fic­a­tion of the work by Maslov and Arnold on the to­po­logy of Lag­rangi­an man­i­folds. In his 1967 work [e1], Arnold proved that the 1-co­homo­logy class of these man­i­folds, con­sidered pre­vi­ously by Maslov as the ob­struc­tion to the quas­i­clas­sic quant­iz­a­tion, can be de­duced by a kind of Gauss map from some uni­ver­sal space, the Lag­rangi­an Grass­man­ni­an \( U/O \), thus be­hav­ing like a char­ac­ter­ist­ic class of usu­al man­i­folds. This class was ori­gin­ally defined as Poin­caré dual to the en­tire set of crit­ic­al points of the Lag­range pro­jec­tion, but turned out to be an in­teger co­homo­logy class un­like the first Stiefel–Whit­ney class defined in the same way for a gen­er­ic map \( M^n \to {\mathbb R}^n \) of an or­din­ary man­i­fold. Look­ing for gen­er­al­iz­a­tions of this work, Arnold posed the prob­lem to study the co­homo­logy classes dual to the sets, where the rank of Lag­rangi­an pro­jec­tions drops by at least \( k \), in par­tic­u­lar to de­cide wheth­er these classes also are well-defined as in­teger co­homo­logy classes. Then Fuchs came to the scene. Firstly, he ex­plained that Lag­rangi­an Grass­man­ni­ans \( U(n)/O(n) \) are not a new ob­ject in to­po­logy: their co­homo­logy rings were cal­cu­lated by A. Borel in the 1950s, and the stable ver­sion of them ap­pears as one of eight spaces in the real Bott peri­od­icity. Fur­ther, he in­deed real­ized the mul­ti­plic­at­ive gen­er­at­ors of their weak in­teger co­homo­logy rings in terms of sets of Lag­rangi­an planes in a non­gen­er­al po­s­i­tion (thus provid­ing an ana­logue of the Schubert cell real­iz­a­tion of Pontry­agin classes in the Lag­rangi­an situ­ation), and showed how the Arnold classes ap­pear as their products.

To­po­lo­gic­al ques­tions arose in the sem­in­ar from three main sources. Firstly, Arnold re­ceived many let­ters and re­prints from col­leagues from around the world. The new ideas and ar­gu­ments in them were for­mu­lated in the terms stand­ard for oth­er sci­entif­ic schools, thus de­mand­ing a trans­la­tion and ad­apt­a­tion; some­times they in­cluded some to­po­lo­gic­al no­tions not stand­ard for the sem­in­ar. Secondly, Arnold him­self of­ten ex­cav­ated a new to­po­lo­gic­al prob­lem or ap­proach from his stud­ies in oth­er areas. As a rule, in this case he solved or ap­plied this con­sid­er­a­tion in the simplest (one-di­men­sion­al, say) ex­ample, and posed the prob­lem to ex­tend it to more ad­vanced situ­ations, re­quir­ing deep and spe­cial to­po­lo­gic­al tech­niques. Fi­nally, the to­po­lo­gic­al is­sues ap­peared in the par­ti­cipant’s talks. In all these cases Fuchs provided us with com­plete, il­lu­min­at­ing in­form­a­tion. The most im­port­ant in this ser­vice was not only his own top-level lit­er­acy in the top­ic (there are enough highly trained to­po­lo­gists of nar­row fo­cus!) but his abil­ity to un­der­stand in­stantly the prob­lem from a dif­fer­ent area, and to re­cog­nize the to­po­lo­gic­al heart of it.

Here are only two ex­amples con­cern­ing my­self (in fact there were more of them). Once Arnold asked me to un­der­stand and de­liv­er in the sem­in­ar a talk on one of S. Smale’s works, in which the co­homo­logy of braid groups was ap­plied to the prob­lems of com­pu­ta­tion­al com­plex­ity. So, I star­ted to talk proudly, how I man­aged to im­prove Smale’s log­ar­ithmic lower bound to a lin­ear one (with coef­fi­cient 0.5), but Fuchs said im­me­di­ately: “Oh, this [Smale’s no­tion in­tro­duced for at­tack­ing this prob­lem] is the A. S. Schwarz genus!” — and then ex­plained where to read about it. I did it and im­proved the coef­fi­cient to al­most 1 (which is also the up­per bound), but it hardly would have been pos­sible if I hadn’t also pre­vi­ously read Fuchs’ work [2]. In an­oth­er case, I star­ted to ex­plain (equally proudly) how I man­aged to un­tie the ho­mo­logy groups of dis­crim­in­ant vari­et­ies to solve one of Arnold’s prob­lem, and Fuchs re­acted: “Oh, this trick is known to the sci­ence, it is called the sim­pli­cial res­ol­u­tion, and is con­veni­ently real­ized by a geo­met­ric con­struc­tion rather than this com­bin­at­or­i­al one.” And in­deed, the com­bin­a­tion of the geo­met­ric and com­bin­at­or­i­al views al­lowed me then to ex­tend this ap­proach to many oth­er situ­ations.

But the main im­pact of Fuchs’ work at the sem­in­ar was not the solu­tion of con­crete prob­lems, nor the good ad­vice. It was the demon­stra­tion of what mod­ern to­po­logy ac­tu­ally is, its lan­guage and philo­sophy, the stand­ard and most ef­fi­cient tools, how un­ex­pec­tedly much can be cal­cu­lated and proved by these tools, which stand­ard mis­takes should be avoided, etc. Ac­tu­ally, the en­tire Arnold sem­in­ar served this pur­pose with re­spect to math­em­at­ics as a whole, but its to­po­lo­gic­al dir­ec­tion was the ser­vice of Dmitry Bor­iso­vich.