LECTURE 12: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES

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1 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES I this lecture we will study the (homological) degree of self-maps of spheres, a otio which geeralizes the usual degree of a polyomial. We will study may examples, establish basic properties of the degree, ad discuss some of the typical applicatios. We will also see how the boudary operator of the cellular chai complex of a space ca be defied i terms of the degrees of self-maps of the spheres. 1. Degrees of maps betwee spheres Let us recall from Lecture 6 that for each 1 we have isomorphisms { H k (D, S 1 ) = H k (S ) = H k (S, ) Z, k = = 0, otherwise. Geerators of the respective free abelia groups of rak oe are fudametal classes or orietatio classes. ote that these are well-defied up to a sig ad we will ow make coheret choices which will the be deoted by [D, S 1 ] H (D, S 1 ), [S ] H (S ) or [S ] H (S, ) respectively. ow, the -sphere is obtaied by gluig the orth hemisphere D ad the south hemisphere DS alog their commo boudary (the equator ). Both hemispheres are just copies of the uit ball D ad as such homeomorphic to. To be more specific, we take the homeomorphism σ : D which is essetially give by a rescalig: D is homeomorphic by a traslatio ad a rescalig to a disc of dimesio cetered at the baryceter of ad with the radius chose such that all vertices of lie o the boudary of that disc; give this disc the we choose σ to be just the obvious homeomorphism give by rescalig. Usig these homeomorphisms we obtai sigular -simplices σ : σ D = D S ad σ S : σ D = D S S, where the udecorated homeomorphisms are obtaied by projectio ito R {0}. Oe ca check that the formal differece z = σ S σ Z (S ) is a cycle which actually represets a geerator [S ] H (S ). Uder the above isomorphisms this also defies the fudametal classes [D, S 1 ] H (D, S 1 ) ad [S ] H (S, ). Recall from Lecture 2 the defiitio of the Hurewicz homomorphism: for every poited space (X, x 0 ) there is a atural group homomorphism h: π 1 (X, x 0 ) H 1 (X). Give a homotopy class α π 1 (X, x 0 ) represeted by a loop γ : S 1 X, h(α) H 1 (X) is the well-defied homology class represeted by the cycle γ e: 1 1 / 1 = S 1 X where e: 1 1 / 1 deotes the quotiet map. The fudametal class [S 1 ] which we costructed above boils dow to edowig S 1 with the usual couter-clockwise orietatio. The 1

2 2 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES quotiet map e: 1 S 1 ca be chose to be the cocateatio e = σ S σ 1. Cosiderig e as a 1-cycle i S 1 ad usig Lemma 3 of Lecture 2 we see that e is homologous to z 1 : e = σ S σ 1 σ S + σ 1 σ S σ = z 1 Thus, the Hurewicz homomorphism associated to a poited space (X, x 0 ) is give by h: π 1 (X, x 0 ) H 1 (X): α α ([S 1 ]) where α : H 1 (S 1 ) H 1 (X) deotes the iduced map i homology. Of course, the homotopy ivariace of sigular homology motivated us to write α (o matter which represetig loop we choose we get the same map i homology!). This descriptio of the Hurewicz homomorphism suggests a extesio to higher dimesios. Give a poited homotopy class α of maps (S, ) (X, x 0 ), we obtai the homology class α ([S ]) H (X). I order to give a precise defiitio of this higher dimesioal Hurewicz homomorphism, oe has first to itroduce higher dimesioal aalogues of the fudametal group. This is doe i ay stadard course o Homotopy Theory, ad the thusly defied Hurewicz homomorphisms do play a importat role. Defiitio 1. Let f : S S be a map. The uique iteger deg(f) Z such that is called the degree of f : S S. f ([S ]) = deg(f) [S ] H (S ) I this defiitio we used of course that H (S ) = Z so that every self-map of H (S ) is give by multiplicatio by a iteger. ote that the defiitio of the degree is idepedet of the actual choice of fudametal classes: a differet choice would amout to replacig [S ] by [S ] ad hece would give rise to the same value for deg(f). Lemma 2. (1) If f, g : S S are homotopic, the deg(f) = deg(g). (2) For maps g, f : S S we have deg(g f) = deg(g) deg(f) ad deg(id S ) = 1. Proof. This is immediate from the defiitio. The secod statemet of this lemma is referred to by sayig that the degree is multiplicative. Example 3. (1) Let f : S 1 S 1 : (x 0, x 1 ) = ( x 0, x 1 ) be the reflectio i the axis x 0 = 0. The deg(f) = 1. Oe way to see this is as follows. Let σ W, σ E : 1 S 1 be paths from the south pole to the orth pole i the clockwise ad the couterclockwise sese respectively. Obviously σ W σ E is a cycle ad it ca be checked (usig a mior variat of Lemma 4 of Lecture 2: see the exercises) that [σ W σ E ] = [S 1 ] H 1 (S 1 ). I particular, we ca use [σ W σ E ] to calculate degrees. Sice f ([σ W σ E ]) = [σ E σ W ] we deduce deg(f) = 1. It follows that a map S 1 S 1 give by a reflectio i a arbitrary lie through the origi has degree 1. (2) Let a: S 1 S 1 : (x 0, x 1 ) ( x 0, x 1 ) be the atipodal map. The deg(a) = 1. This follows from the previous example ad the multiplicativity of the degree sice the atipodal map is the compositio of two reflectios. (3) Let τ : S 1 S 1 be give by τ(x 0, x 1 ) = (x 1, x 0 ). The deg(τ) = 1. Ideed, τ is a reflectio i a lie.

3 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES 3 I order to exted these examples to higher dimesios, let us recall that we ca costruct S +1 as the suspesio of S, S +1 = S(S ). If we write S = {(x 0,..., x ) R +1 x 2 i = 1} the S+1 = S(S ) is homeomorphic to the quotiet of S [ 1, 1] obtaied by idetifyig all (x 0,..., x, 1) to a poit, the orth pole, ad all (x 0,..., x, 1) to a poit S, the south pole. The homeomorphism is iduced by the map S [ 1, 1] S +1 : ((x 0,..., x ), t) (x 0,..., x, t) where x i = 1 t 2 x i. Uder this isomorphism, it is clear that the suspesio of f : S S i the stadard coordiates for S +1 is Sf : S +1 S +1 : (x 0,..., x +1 ) (rf(r 1 x 0,..., r 1 x ), x +1 ) where r = 1 x I particular, Sf restricts to f o the meridia x +1 = 0. Propositio 4. For ay f : S S ad the associated Sf : S +1 S +1, we have a equality of degrees deg(sf) = deg(f). Proof. Write S +1 = S(S ) = A B for the orther ad souther hemispheres A ad B (give by x +1 0, resp. x +1 0). The A B = S is the meridia. Sice A ad B are cotractible, the Mayer-Vietoris sequece (for slight extesios to ope eighborhoods of A ad B which are homotopy equivalet to A ad B) gives: H +1 (S +1 ) = H (S ) (Sf) f H +1 (S +1 ) = H (S ) The square commutes by aturality of the Mayer-Vietoris sequece. From this, the statemet follows by tracig the geerator [S +1 ] through this diagram. I more detail, sice is a isomorphism, we have ([S +1 ]) = ɛ [S ] with ɛ { 1, +1}, ad hece (f )([S +1 ]) = f (ɛ [S ]) = ɛ f ([S ]) = ɛ deg(f) [S ]. Similarly, if we trace [S +1 ] through the lower left corer, we calculate ( (Sf) )([S +1 ]) = (deg(sf) [S +1 ]) = deg(sf) ([S +1 ]) = deg(sf) ɛ [S ]. Comparig these two expressios cocludes the proof. This proof is a further istace of a calculatio showig that aturality of certai log exact sequeces is ot just a techical issue but actually useful i calculatios. With this preparatio we ow obtai the followig higher-dimesioal versios of Example 3. Propositio 5. (1) The degree of the map f : S S : (x 0, x 1,..., x ) ( x 0, x 1,..., x ) is 1. More geerally, the degree of the ay reflectio at a arbitrary hyperplae through the origi is 1. (2) Let f : S S be give by f(x 0, x 1,..., x ) = (ɛ 0 x 0, ɛ 1 x 1,..., ɛ x ) for some sigs ɛ i. The deg(f) = ɛ 0... ɛ. I particular, if f = a is the atipodal map (thus all ɛ i are 1) the deg(a) = ( 1) +1. (3) Let f : S S be give by f(x 0, x 1,..., x ) = (x 1, x 0, x 2, x 3,..., x ). The deg(f) = 1.

4 4 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES (4) Let f : S S : (x 0,..., x ) (x τ(0),..., x τ() ) for some permutatio τ Σ +1. The deg(f) = sg(f) is the sigature of the permutatio τ. Proof. Part (1) follows Example 3.(1) ad Propositio 4 together with the observatio that all reflectios at hyperplaes are homotopic. The remaiig parts are a immediate cosequece of (1) ad the multiplicativity of the degree (to obtai (4), write a arbitrary permutatio as a sequece of traspositios). Lemma 6. Let f, g : X S R +1. If f(x) g(x) for all x X the f g. Proof. Let H : X [0, 1] R +1 be give by H(x, t) = (1 t)f(x) + tg(x). The for a fixed x, the partial map H(x, ): [0, 1] R +1 is the lie from f(x) to g(x). By assumptio, this lie does ot pass through the origi, so we ca ormalize H to obtai the map K : X [0, 1] S : (x, t) H(x, t)/ H(x, t), a well-defied homotopy from f to g. Corollary 7. Let f : S S. (1) If f has o fixed poit, the deg(f) = ( 1) +1. (2) If f has o atipodal poit (a poit x with f(x) = x), the deg(f) = 1. Proof. For the first statemet, if f(x) x for all x, the f is homotopic to the atipodal map a defied by a(x) = x, accordig to the lemma. But sice the degree is homotopy-ivariat we ca coclude by Propositio 5.(2). The proof of the secod statemet is similar, sice by the lemma, f(x) x for all x implies that f is homotopic to the idetity. Corollary 8. If is eve, the ay f : S S has a fixed poit or a atipodal poit. Proof. If f has either a fixed poit or a atipodal poit, the, by the previous corollary, the degree of f has to be 1 ad 1 which is impossible. Corollary 9. Let be eve, the ay vector field v o S has a zero. Proof. Such a vector field assigs to ay x S a vector v(x) based at x ad lyig i the hyperplae taget to S R +1 at x. If v(x) 0 for all x, the the map v = v/ v : S S (obtaied by ormalizig the vector field ad cosiderig the vectors as attached to the origi of R +1 ) is such that each v(x) is a uit vector parallel to the hyperplae taget to S at x. But this cotradicts the previous corollary, sice v would have either a fixed poit or a atipodal poit. For = 2 this is sometimes referred to as the hairy ball theorem: you caot comb a hairy ball without a partig. The corollary tells us that there are o o-where vaishig vector fields o eve-dimesioal spheres. Thus oe might woder what happes for odd-dimesioal spheres. It is easy to costruct a o-where vaishig vector field o S 2+1, v(x 0, x 1,... x 2, x 2+1 ) = ( x 1, x 0, x 2, x 3,..., x 2+1, x 2 ). For a log time it was a ope problem i algebraic topology to determie the maximal umber of everywhere liearly idepedet vector field o spheres, which was fially solved by Adams usig fairly advaced techiques. Usig sigular homology we maaged to obtai first partial results i that directio.

5 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES 5 2. The cellular boudary operator If we wat to be able to calculate the cellular homology of CW complexes which have cells i subsequet dimesios, the it is helpful to have a more geometric descriptio of the cellular boudary homomorphism. Such a descriptio ca be obtaied by meas of the homological degrees of self-maps of the spheres, as discussed i Sectio 1. Let us recall that the cellular chai groups are free abelia groups, i.e., we have isomorphisms J Z = C cell (X) where J deotes the idex set for the -cells of X. Uder these isomorphisms, the cellular boudary maps hece correspod to homomorphisms J Z = C cell This compositio seds every -cell σ of X to a sum (X) cell C 1(X) cell = Z. J 1 σ Σ τ J 1 z σ,τ τ for suitable iteger coefficiets z σ,τ. To coclude the descriptio of this assigmet we thus have to specify these coefficiets. For that purpose, let us fix a -cell σ ad a ( 1)-cell τ. The -cell σ comes with a attachig map χ σ X ( 1) S 1 D X (). ow, associated to this attachig map we ca cosider the followig compositio 1 χσ f σ,τ : S X ( 1) X ( 1) /X ( 2) = S 1 S 1 J 1 i which the last arrow maps all copies of the spheres costatly to the base poit except the oe belogig to the idex τ J 1 o which the map is the idetity. Thus, for each such pair of cells we obtai a poited self-map f σ,τ of S 1 ad its degree turs out to coicide with z σ,τ. ote that sice S 1 is compact it follows that for ay -cell σ there are oly fiitely may ( 1)-cells τ such that f σ,τ is ot the costat map. Thus, the sums i the ext propositio are well-defied. Workig out the details, by usig the log exact sequece of the homology of the pair, oe ca easily verify the followig result: Propositio 10. Uder the above isomorphisms the cellular boudary homomorphism is give by the map Z Z: σ Σ τ J 1 deg(f σ,τ )τ. J 1 J Hece i the cotext of a specific CW complex i which we happe to be able to calculate all the degrees showig up i the propositio, the problem of calculatig the homology of the CW complex is reduced to a purely algebraic problem. Let us give a brief discussio the example of the real projective spaces RP, 0. We begi by recallig that RP is obtaied from S by idetifyig atipodal poits. Hece, there are quotiet maps p = p : S RP. The real projective space RP ca be edowed with a CW structure such that there is a uique k-cell i each dimesio 0 k. Oe ca check that the cellular boudary homomorphism : Ck cell (RP ) Ck 1 cell (RP ), 0 < k is zero if k is odd ad multiplicatio by 2 if k is eve. From this oe ca derive the followig calculatio.

6 6 LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES Example 11. The homology of a eve-dimesioal real projective space is give by H k (RP 2m ) Z, k = 0, = Z/2Z, k odd, 0 < k < 2m 0, otherwise. The homology of odd- I particular, the top-dimesioal homology group H 2m (RP 2m ) is zero. dimesioal real projective spaces looks differetly ad is give by H k (RP 2m+1 ) Z, k = 0, 2m + 1 = Z/2Z, k odd, 0 < k < 2m + 1 0, otherwise. I this case, the top-dimesioal homology group is agai simply a copy of the itegers. geerator of this group is called fudametal class of RP 2m+1. ote that these are our first examples of spaces i which the homology groups have o-trivial torsio elemets. This should ot be cosidered as somethig exotic but istead it is a geeral pheomeo. We coclude this lecture with a short outlook. There is a axiomatic approach to homology which is due to Eileberg ad Steerod. By defiitio a homology theory cosists of fuctors h, 0, from the category of pairs of topological spaces to abelia groups together with atural trasformatios (called coectig homomorphisms) δ : h (X, A) h 1 (A, ), 1. This data has to satisfy the log exact sequece axiom, the homotopy axiom, the excisio axiom, ad the dimesio axiom. We let you guess the precise form of the first three axioms, but we wat to be specific about the dimesio axiom. It asks that h k (, ) is trivial i positive dimesios. Thus, the oly possibly o-trivial homology group of the poit sits i degree zero ad that group h 0 (, ) is referred to as the group of coefficiets of the homology theory. So, parts of this course ca be summarized by sayig that sigular homology theory defies a homology theory i the sese of Eileberg-Steerod with itegral coefficiets. I the sequel to this course we study closely related algebraic ivariats of spaces, amely homology groups with coefficiets ad cohomology groups. Ay