1^ 



648 



RKPOUT — 1884. 



''ii 



( -.]: 



11 



III 



ft 



ft -Si 



M 





Mi 



D^^\„.„.-:rAH;:A^TA-i?A''S 



+ /,A^AS^-^,A^ST. 



Those exprpssioiis liavo to lie iiitnxluct'd in Jinosclil's <liscii:niii;nit, ar,>l si will 

 f:lve iramediatt'ly the tact Invariant of^* - and a ;' for whk'h we have heoi liokinf^. 



G. On the '■ Anal ij sis Sit us ^ of Tli rei'd i nicn-iunial Sp.icC'i. 

 Bij Professor Waltiier Dvck, B.i'h. 



The followiiinf considorations refer to ,he analysis situs of tlin't'dimcinsional 

 spaces, and its formulatiou is called forth by certain lesearches on the theory of 

 functions, which, however, I canot enter upon iiere. 



The object is to determine certain charaeteristical uunibers for closed throe- 

 iliraensional spaces, analojjfous to those introduced by liitsmann in the tlieory of 

 his surfaces, so that their indentity ahows the possibility of its ' one-one {geometri- 

 cal correspondence.' 



yupposinj,' every ptart of the space in question liehaves itself as our ordinary 

 euclidian space, with this restriction, that the iniinitely distant points are to be 

 considered as condensed at one sinjrle point (' Kaiun der reciproken Itadien '). 

 Collecting nnder one representative all those S])aces, betweeji which a one-one cor- 

 respondence is possible, we can form all post-ible closed threedimensi(jnal spaces by 

 the foUowinjr procedure: 



We cut out of our space 21; parts, limited by closed surfaces, each pair beiiii;- 

 respectively of the deliciency (Geschlecht) ^>,,^a,, . . ,p. Then, by establishiiijr 



a mutual one-one correspondence between every two surfaces, we close the spacu 

 tlius obtained, 'i'he numbers^),, j).,, . , . p ot the surfaces, thus made use of, and 



the manner of tlieir mutual correspondence then form what we may call the dis- 

 tinctive characteristic of our space. This cimracteristic is determined : — 



1. ]Jy the existence of certain closed surfaces, which are not able to isolate a pari 

 < if the space. These are surfaces surrounding the above-named surfaces of the deli- 

 ciency;;,,;), , . . p^. 



2. IJy t!ie existence of certain closed curves in our space, which can n'.'ither Ijc 

 transformed into each other, nor be drawn together into one point. 



We will now consider the last-named characteristic, whii'h, so far as I know, 

 has not been elsewhere discussed, l^et me explain it by an easy examjile, suitable 

 to show^ the general particularities. Suppose two rings (of tiie detlciency 1) cut 

 oft' of our ordinary space. According to the manner in which the one-one corre- 

 spondence of these two surfaces is delined, essentially ditlering spaces are formed. 

 First, for example, we can make them correspond so that meridian curves fall nii 

 meridian curves, and latitudinal curves into latitudinal curves. Then there exist 

 curves which cannot be contracted into one point. For if we put a closed curvi^ 

 surrounding the first ring, the curve, by all expansions and deformations it is liable 

 to, always encloses one of the two rings. On the contrary, supposing we had madtf 

 the meridional curves to correspond to the latitudinal ones and vice versa, curves 

 of the above description would not have been found. For a curve surrounding the 

 one ring can lirst bo contracted iato a meridional curve of this ring, 'i'his curve 

 is identical with a latitudinal curve of the second ring, and this last-mentioned 

 curve can bo removed from the ring into our space, and therefore be contracted 

 into a point. 



Ill this way the particular correspondences, above described, lietwoen every two 

 of our surfaces give rise to particular kinds of closed spaces. The enumeration of 

 these spaces is immediately connected with the enumeration of the canonical 

 orthogonal suKstitutions, which give rise to 2p new periods from the periods 



«, 



w.,p in the theory of Abelian integrals, according to Kronecker. 

 1 hope to develope this subject further on another occasion. 



