Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks. Let S j dente the bundary f E j, and let F = D n j IntE j. Then the fllwing hld: (i) The hmlgy grups H q (F ) are zer if q 0, n 1, and H 0 (F ) = Z. (ii) In the remaining dimensin we have H n 1 (F ) = Z k, and the inclusin induced mappings H n 1 (S j ) H n 1 (F ) send generatrs f the dmains int a set f free generatrs fr the cdmain. (iii) If we define standard generatrs fr H n 1 (S j ) by taking the images f the standard generatr fr H n 1 (S n 1 ) under the cannial hmemrphims S n 1 S j, then the image f the geeratr fr H n 1 (S n 1 ) in H n 1 (F ) is equal t the sums f the standard free generatrs fr H n 1 (F ). Gemetrically, F is a disk with k hles; a picture f ne example is included n the last page f this dcument. The standard hmemrphisms S n 1 S j arise frm the hmemrphisms frm R n t itself which send v R n t (r j v) + p j, where r j is the radius f E j. Prf f the therem The first step is t replace F by an pen set. CLAIM 1. The clsed set F is a strng defrmatin retract f R n {p 1,, p k }. Prf f Claim 1. Observe that R n {p 1,, p k } is the unin f F with the punctured clsed disks E j {p j }, and the intersectin is j S j. Therefre it is enugh t shw that fr each j the sphere S j is a strng defrmatin retract f E j {p j }. We can cnstruct the retractins E j {p j } S j by the standard frmula ρ j (v) = p j + r j v p j (v p j) fr pushing pints int the bundary radially, and if ϕ j : S j E j {p j } is the inclusin mapping then ϕ ρ j is hmtpic t the identity by a straight line hmtpy. CLAIM 2. D n. Prf f Claim 2. disk D n R n. If D R n is a clsed metric disk, then (R n ) D is hmemrphic t the interir f The first step is t reduce the prf t the familiar case where D is the unit If the result is true when D = D n, then it is als true fr every clsed disk E centered at the rigin, fr the hmemrphism M r : R n R n defined by v r v (where v > 0) sends D n t the disk f radius r, and fr general reasns it extends t a hmemrphism f the ne pint cmpactificatin (R n ). If r is the radius f E, then this hmemrphism sends D n t E and hence als sends the cmplement f D n hmemrphically t the cmplement f E. Next, if the result is true fr every clsed disk E centered at the rigin, then it is true fr all clsed metric disks D, fr if p is the center f D, then T (v) = v + p is a hmemrphism f R n t itself and hence extends t a hmemrphism frm (R n ) t itself. If we chse E t have the 1
same radius as D, then this hmemrphism maps D t E and hence als maps the cmplement f D t the cmplement f E. Finally, we have t shw the result is true fr D n. Let G R n be the set f all vectrs v such that v 1. We claim that G is hmemrphic t D n {0}; an explicit hmemrphism is given by sending v t v 2 v, s that the image f v pints in the same directin but has length v 1. If we extend this hmemrphism t ne pint cmpactificatins and nte that the ne pint cmpactificatin f D n {0} is hmemrphic t D n, we btain a hmemrphism frm G t D n such that the unit sphere is sent t itself. Taking cmplements f the unit sphere, we see that G S n 1 = (R n ) D n is hmemrphic t the interir f D n. STEP 3. Cmputatin f H (R n {p 1,, p k }) fr n 2. Let U j dente the interir f the disk E j. Then by excisin we have H (R n, R n {p 1,, p k }) = H ( j U j, j U j {p j }) = j H (U j, U j {p j }) where the secnd ismrphism hlds because the hmlgy f a space splits int the direct sum f the hmlgy grups f its arc cmpnents. These relative grups are Z in dimensin n and zer therwise, and since n 2 the result in this case fllws because the lng exact hmlgy sequence f (R n, R n {p 1,, p k }) yields ismrphisms frm H q+1 (R n, R n {p 1,, p k }) t H q (R n {p 1,, p k }) if q > 0 and frm H 0 f the latter t H 0 (R n ) = Z. Befre prceeding t the final step, we shall discuss the cnstructin f cannical generatrs in mre detail. Fr ur purpses it will suffice t begin by taking a cannical generatr fr H n 1 (bbr n, R n = {0}) = Z; there are ways f chsing such generatrs fr all values f n cannically, but we shall nt try t explain hw this can be dne. Cnsider the fllwing cmmutative diagram, in which p Int D n and we identify S n with the ne pint cmpactificatin f R n : H q 1 (S n {p}) H q 1 (S n D n ) H q 1 (S n {0}) H q (S n, S n {p}) H q (S n, S n D n ) H q (S n, S n {0}) H q (R n, R n {p}) H q (R n, R n D n ) H q (R n, R n {0}) By Claim 2 we knw that S n D n is cntractible, and we als knw that S n {p} is cntractible fr all p in the interir f D n by the symmetry prperties f S n and the fact that S n {v} = R n if v is the pint at infinity. Therefre the hmmrphisms in the first rw f the diagram are ismrphisms. Next, the vertical arrws frm the secnd rw t the first are the bundary hmmrphisms in lng exact sequences f pairs, and therefre a Five Lemma argument shws that the hmmrphisms in the secnd rw are als ismrphisms. Finally, the vertical arrws frm the third rw t the secnd are excisin ismrphisms, and therefre the hmmrphisms in the third rw are als ismrphisms. We can then use the third rw t define a cannical generatr fr H n (R n, R n {p}) by taking the class crrespnding t the chsen generatr fr H n (R n, R n {0}). STEP 4. Cmputatin f the image f H n (R n, R n {0}) = Z in H n (R n, R n {p 1,, p k }) = Z k fr n 2. 2
By the splitting result mentined earlier, it suffices t cnsider the maps H n (R n, R n {0}) H n (R n, R n {p j }) fr each j, and the preceding discussin shws that these maps are ismrphisms which preserve cannical generatrs. Therefre the image f H n (R n, R n {0}) = Z in H n (R n, R n {p 1,, p k }) = Z k is merely the sum f the cannical free generatrs r the cdmain. STEP 5. Cmputatin f the image f H n 1 (S n 1 ) = Z in H n 1 (F ) = Z k fr n 2. Fr each j such that 1 j k, let F j = D n U j, where U j is the small pen disk centered at p j. We shall begin by analyzing a cmmutative diagram which is related t the previus ne: H n (R n, R n {p 1,, p k }) = = j H n (R n, R n {p j }) = H n 1 (R n {p 1,, p k }) j H n 1 (R n {p j }) = = H n 1 (F ) j H n 1 (F j ) The arrw in the first rw is an ismrphism by excisin, the arrws frm the first t secnd rws are ismrphisms by the lng exact hmlgy sequences fr the pairs (the adjacent terms in each case are psitive dimensinal hmlgy grups f R n ), and the arrws frm the third t secnd rws are ismrphisms by Step 1. By the direct sum decmpsitins n the right, it suffices t analyze the image in hmlgy when we are nly remving the pint p j r the pen disk U j centered at p j, where 1 j k. At this pint we need t be careful abut chsing the right signs fr ur free generatrs f hmlgy grups, especially in view f the applicatin we have in mind. If Σ is a sphere f radius r centered at p R n, we take the hmemrphism S n 1 Σ cnstructed in Step 2: First stretch r shrink the sphere S n 1 f radius 1 centered at 0 t a cncentric sphere f radius r, and then map this t the crrespnding sphere centered at p via the translatin v v +p. Then by the cmments in the preceding paragraph we shall have prved the prpsitin if we can shw the fllwing: Let S 1 R n be the sphere f radius a centered at p, suppse that the disk it bunds is cntained in the interir f the disk f radius b centered at sme pint q, and let S 2 dente the bundary sphere f that disk. Let f 1 : S n 1 S 1 and f : S n 1 S 2 be the hmemrphisms given as abve. Chse a generatr ω f H n 1 (S n 1 ). Then the images f f 1 (ω) and f 2 (ω) in R n {p} are equal. T prve this, let S 3 be the sphere f radius b centered at p, and let j 1 and j 3 dente the inclusins int R n = {p}. Then j 3 f 3 j 1 f 1 by the radial stretching hmtpy sending (x, t) t (1 t) f 1 (x) + t f 3 (x). Therefre f 1 (ω) = f 3 (ω). By definitin we als have f 2 (x) = f 3 (x) + q p; if j 2 is th inclusin f S 2 in R n {p}, it will suffice t prve that j 3 f 2 j 2 f 2, and this will fllw if the image f the straight line hmtpy H(x, t) = f 3 (x) + (1 t)(q p) is cntained in R n {p}. Since q p = d and a is the radius f the disk bunded by S 1, the cnditin that ne sphere is cntained in the interir f the pen disk bunded by the ther means that d + a < b (Prf: If w is chsen s that w p is a negative multiple f q p and w p = a, then w S 1, s that w is als in the pen disk bunded by S 2 and therefre b > w q = w p + q p = a + d). We need t shw that x = b implies that H(x, t) p, r equivalently that x = b implies that H(x, t) p > 0. But we have H(x, t) p = f 3 (x) p + (1 t) (q p) 3
which means that if 0 t 1 (hence als 0 1 t 1) then H(x, t) p f 3 (x) p (1 t) q p > b (1 t) d which is what we wanted t prve. b d > a > 0 A degree frmula We shall use the therem t prve an abstract, multidimensinal versin f a result which plays a key rle in cmplex analysis when n = 2. COROLLARY. Let F be as in the therem, and suppse that we are given a cntinuus mapping g : F R n {0}, and fr each j let E j be the subdisk in D n whse interir is remved t frm F. Let h j : S n 1 F be the cmpsite f the standard hmemrphism S n 1 Bdy E j with the inclusin f Bdy E j in F. Then we have a summatin frmula deg (g S n 1 ) = j deg (g h j ). Sectin 4.5 f Ahlfrs, Cmplex Analysis (Third Editin), describes implicatns f this result fr cmplex functin thery. Prf f the crllary. Let ω be the standard generatr f H n 1 (S n 1 ) described in Step 5, and let h 0 : S n 1 F be the inclusin mapping. Then by the therem we have h 0 (ω) = j h j (ω) and if we apply g t bth sides we btain a similar identity with h j replaced by g h j = (g h j ) fr all j. By the definitin f degree we knw that the image f ω under the latter map is equal t the degree f g h j times ω if j > 0, and if h = 0 then the image f ω is equal t the degree f g S n 1 times ω. 4
Drawing f a disk with hles The fllwing is a picture f a typical set F satisfying the cnditins in the main therem. Nte that the hles may be irregularly distributed thrughut the disk and that the radii f the hles may als differ. Als, in general the center f the circle might nt be ne f the deleted center pints. In the 2 dimensinal case, the main therem implies that uter circle with a cunterclckwise parametrizatin is hmlgus t the sum f the inner circles with cunterclckwise parametrizatins (in ther wrds, the tw 1 dimensinal cnfiguratins determine the same hmlgy class).