ALGEBRAIC TOPOLOGY NOTES WEEK 2 Cellular Boundary Formula Let be a complex with cells n in each dimension . Recall that if is a X CW feαgα n X CW complex we can build a cellular chain complex (C∗; d∗) where C∗ = Hn(Xn;Xn−1) = # of n cells To describe the cellular boundary map it suces to say what it does to the Z : dn generators of the chain group Cn: To do this, we choose the natural basis for Cn which assigns to each n a generator of and to each n−1 a generator of It follows from linearity eα Cn eβ Cn−1: that n P n−1 for some integers Our rst theorem will give a convenient dn(eα) = β dαβeβ dαβ: representation to the coecients dαβ: Theorem. The coecients in n P n−1 may be computed as the degree of dαβ dn(eα) = β dαβeβ 'n the map n−1 α n−1 given by attaching the cell n to via the attaching Sα / Xn−1 / Sβ eα Xn−1 map n from the CW structure on and then collapsing ` n−1 to a point. 'α X Xn−2 γ6=β eγ Proof. We will proceed with the proof by chasing the following diagram, where the map ∆αβ∗ is dened so that the top right square commutes. ∆ n n−1 @ n−1 αβ∗ n−1 Hn(Dα;Sα ) ' / Hn(Sα ) / Hn−1(Sβ ) O n n−1 q Φ∗ 'α∗ β∗ @n q∗ ~ ~ n−1 n−1 Cn(X) / Hn−1(Xn−1) / Hn−1(Xn−1=Xn−2) = ⊕βHn−1(eβ =@eβ ) OO OO d OOO n n−1 OO j∗ ' OOO OO' ' Xn−1 Xn−2 C (X) / Hn( ; ) n−1 Xn−2 Xn−2 Recall that our goal is to compute n . The upper left square is natural and therefore dn(eα) ∗ ∗−1 commutes (it is induced by the map Φ:(D ;S ) ! (X∗;X∗−1), see Hatcher p.127), while the lower left triangle is part of the exact diagram dening the chain complex C∗(X) and is dened to commute as well. Appealing to naturality, the map gives a unique n so that Φ Dα n n n. Since the top left square and the bottom left triangle both commute, this Φ (Dα) = eα gives that n n−1 n n . dn(eα) = j∗ ◦ 'α∗ ◦ @(Dα) Looking to the bottom right square, recall that since X is a CW complex, (Xn;Xn−1) is a ~ good pair. This gives the isomorphism Cn−1(X) = Hn−1(Xn−1;Xn−2) ' Hn−1(Xn−1=Xn−2) ~ But, we similarly have Hn−1(Xn−1=Xn−2) ' Hn−1(Xn−1=Xn−2;Xn−2=Xn−2), where the iso- morphism is induced by the quotient map q collapsing Xn−2 (Hatcher 2.22 p.124). The bottom right square commutes by the denition of n−1 and from which it follows j∗ q∗ that α = n−1 α where formally we should precompose with the isomorphism be- dn(en) q∗'α∗ @Dn ~ tween Cn−1(X) and Hn−1(Xn−1=Xn−2) in the left hand side so that everything is in the same space. This last map takes the generator α to some linear combination of gener- Dn ators in ~ n−1 n−1 . To see which generators it maps to, we project down to ⊕βHn−1(eβ =@eβ ) 1 ALGEBRAIC TOPOLOGY NOTES WEEK 2 2 the summands to obtain α = P n−1 α. As noted before, we have dened β dn(en) β qβ∗q∗'α∗ @Dn n−1. Writing α = P β, we can see from the denition of the ∆αβ∗ = qβ∗q∗'α∗ dn(en) β ∆αβ∗@Dn above maps that is multiplication by . ∆αβ∗ dαβ Computing Cellular Homology Example. Mg (the orientable surface of genus g) We rst claim that we can construct a CW structure on Mg as follows: 2g 0 a 1 a 2 1 2 Mg = fD g fDαg fD g=['α;' ] α=1 where we denote by 1 2 the relations n for n and where the maps 1 are ['α;' ] 'α(x) ∼ x x 2 @Dα 'α the collapsing maps and the map 2 takes the generator of 2 to the word −1 −1 −1 −1 ' D a1b1a1 b1 :::agbgag bg 1 if we write ag and bg as the generators associated to the 2g copies of D : We can justify that this denition agrees with the geometric denition of Mg by writing Mg as the g−fold con- nected sum of the torus T 2 and considering the fundamental polygon of T 2: As an example of this construction in the case g = 2 we can consider two tori as below, where we remove the two red circles and identify their boundaries: Topologically, once we glue the tori together along the identied circles, this gives the double torus (M2) seen below: We can realize this construction at the level of the fundmanetal polygons (which will give rise more naturally to the cell structure above) by considering the fundamental polygon of the torus: ALGEBRAIC TOPOLOGY NOTES WEEK 2 3 The above polygon has a natural CW structure which is exactly what we described above (the polygon above can be viewed as a single vertex identied as the boundary of each of the four edges attached to a two cell in the center where the attaching map is consistent with the identications of the edges). Taking two copies of this fundamental polygon (one for AB and one for CD), we can remove a disk from each and identify the boundary circles E to construct the connected sum. If we cut along a diagonal (the picture below corresponds to cutting the AB square from the bottom left vertex to E and the CD square from the top right vertex to E then gluing them) we can glue two of these squares together to obtain a space topologically equivalent to the above construction. Notice that we may delete all of E except its endpoints in this new picture since it is topologically the same as the area around it. We now compute the cellular chain group C∗(Mg): Since Ck(Mg) ' Hn(Xn;Xn−1) where Xn is the n−skeleton of Mg we have d3 d2 d1 d0 0 / Z / Z2g / Z / 0 ALGEBRAIC TOPOLOGY NOTES WEEK 2 4 We demonstrated in the last class that d1 is the zero map because there is only one one- 2 cell. There is only one two-cell as well, so we only need to nd d2(e ) which we compute via the cellular boundary formula above. Recall from the beginning of this example that the attaching map sends the generator to −1 −1 −1 −1. Consider the summand a1b1a1 b1 :::agbgag bg dea1 in P 1 . When we collapse all the other cells to a point, the word dening the attaching β deβeβ map −1 −1 −1 −1 reduces to −1 . We can see this geometrically in the case a1b1a1 b1 :::agbgag bg a1a1 = 0 g = 2 in the above picture by collapsing all the cells except A to a point, at which point the map is given by 2 By symmetry then it follows that d2(e ) is the zero map. It then follows that the homology groups of Mg are given by 8 i=0,2 <>Z 2g i=1 Hn(Mg) = Z :>0 otherwise Example. Ng (the nonorientable surface of genus g) Formally, 2 2. We construct this directly as above via the labelling scheme Ng = RP #:::#RP which sends a generator to 2 2 which coincides in dimension two ( ) with 2. a1:::ag g = 1 RP The short exact sequence for the cellular chains is now given by d3 d2 d1 d0 0 / Z / Zg / Z / 0 where as before d1 = 0 because there is only one cell in dimension one. Now to compute 2 d2(e ) we again apply the cellular boundary formula. The same collapsing happens in the quo- tient space except now for each since we attach along the map Q 2. It follows then deai = 2 i i ai that . We now choose a basis for g given by d2(e) = (2; :::; 2) Z (1; 0; :::; 0); :::; (0; :::; 0; 1; 0); (1; :::; 1) (that is the standard basis with the last basis vector replaced with (1; :::; 1)). We can then compute directly that 8 i=0 <>Z g−1 i=1 Hn(Ng) = Z ⊕ Z2 :>0 otherwise n Example. RP n We showed in the last notes that we can construct RP as the quotient space of Sn where n we identify antipodal points. RP = Sn= ∼ has one cell in each dimension 0; 1; :::; n where we consider the CW structure of Sn with two cells in each dimension. The attaching map of n k−1 ek in RP is the two-fold cover Sk−1 ! RP . We now consider the long exact sequence for the chain groups dn+1 dn d1 d0 0 / Z / :::Z / Z / 0 ALGEBRAIC TOPOLOGY NOTES WEEK 2 5 Since is a map from k−1 k−1 k−1 k−2 we can compute its degree dk dk : S ! S = RP =RP k−1 k−2 directly by considering any y 2 RP nRP for which we know that (letting ' be the attaching map) '−1(y) = fy; a ◦ yg where a is the antipodal map. This is the usual two-fold cover of Sk−1 and so is a local homeomorphism. We consider a neighborhood V of y and the two neighborhoods U1 and U2 given to exist by the local homeomorphism property.