42. Mon, Dec. 8 Last time, we were discussing CW complexes, and we considered two di erent CW structures on S n. We continue with more examples.

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1 42. Mon, Dec. 8 Last tme, we were dscussng CW complexes, and we consdered two d erent CW structures on S n. We contnue wth more examples. (2) RP n. Let s start wth RP 2. Recall that one model for ths space was as the quotent of D 2, where we mposed the relaton x x on the boundary. If we restrct our attenton to the boundary S, then the resultng quotent s RP, whch s agan a crcle. The quotent map q : S! S s the map that wnds twce around the crcle. In complex coordnates, ths would be z 7! z 2. The above says that we can represent RP 2 as the pushout S D 2 q S RP 2 If we buld the -skeleton S usng a sngle 0-cell and a sngle -cell, then RP 2 has a sngle cell n dmensons apple 2. More generally, we can defne RP n as a quotent of D n by the relaton x x on the boundary S n. Ths quotent space of the boundary was our orgnal defnton of RP n. It follows that we can descrbe RP n as the pushout S n D n q RP n RP n Thus RP n can be bult as a CW complex wth a sngle cell n each dmenson apple n. (3) CP n. Recall that CP = S 2. We can thnk of ths as havng a sngle 0-cell and a sngle 2-cell. We defned CP 2 as the quotent of S 3 by an acton of S (thought of as U()). Let : S 3! CP be the quotent map. What space do we get by attachng a 4-cell to CP by the map? Well, the map s a quotent, so the pushout CP [ D 4 s a quotent of D 4 by the S -acton on the boundary. Now nclude D 4 nto S 5 C 3 va the map q X '(x,x 2,x 3,x 4 )=(x,x 2,x 3,x 4, x 2, 0). (Ths would be a hem-equator.) We have the dagonal U() acton on S 5. But snce any nonzero complex number can be rotated onto the postve x-axs, the mage of ' meets every S -orbt n S 5, and ths ncluson nduces a homeomorphsm on orbt spaces D 4 U() = S 5 U() = CP 2. We have shown that CP 2 has a cell structure wth a sngle 0-cell, 2-cell, and 4-cell. Ths story of course generalzes to show that any CP n can be bult as a CW complex havng a cell n each even dmenson. 43. Wednesday, Dec. 0 (4) (Torus) In general, a product of two CW complexes becomes a CW complex. We wll descrbe ths n the case S S,whereS s bult usng a sngle 0-cell and sngle -cell. Start wth a sngle 0-cell, and attach two -cells. Ths gves S _ S. Now attach a sngle 2-cell to the -skeleton va the attachng map defned as follows. Let us refer to the two crcles n S _ S as ` and r. We then specfy : S! S _ S by `r` r. What we mean s to trace out ` on the frst quarter of the doman, to trace out r on the second 68

2 quarter, to run ` n reverse on the thrd quarter, and fnally to run r n reverse on the fnal quarter. We clam that the resultng CW complex X s the torus. Snce the attachng map : S! S _ S s surjectve, so s D 2 : D 2! X. Even better, t s a quotent map. On the other hand, we also have a quotent map I 2! T 2, and usng the homeomorphsm I 2 = D 2 from before, we can see that the quotent relaton n the two cases agrees. We say that ths homeomorphsm T 2 = X puts a cell structure on the torus. There s a sngle 0-cell (a vertex), two -cells (the two crcles n S _ S ), and a sngle 2-cell. Let s talk about some of the (nce!) topologcal propertes of CW complexes. Lemma 43.. Let E = {e n } be the set of all cells n X. Then X s a quotent of a E D n. In partcular, A X s open (or closed) f and only f, for each cell and correspondng characterstc map : D n! X, the premage (A) s open (or closed) n D n. Proof. The forward mplcaton s clear by contnuty of the '. For the other drecton, suppose that each (A) s open. Then A \ X 0 s open n X 0,snceX 0 s just the dsjont unon of ts cells. Now assume by nducton that A \ X n s open n X n. But, by the constructon of the pushout, the n-skeleton X n s a quotent of X n q ` D n. Snce A \ X n pulls back to an open set n each pece of ths coproduct, t must be open n A \ X n by the defnton of the quotent topology. Now, snce A \ X n s open n X n for all n, A s open n X by property W. Theorem Any CW complex X s normal. Proof. Frst, X s T by the Lemma snce any pont obvously pulls back to a closed subset of every D n. Let A and B be dsjont closed sets n X. We wll show that X s normal by buldng a Urysohn functon f : X! [0, ] wth f(a) 0 and f(b). Because X satsfes property W, a functon f defned on X s contnuous f and only f ts restrcton to each X n s contnuous. We thus buld the functon f by buldng ts restrctons f n to X n. On X 0,wedefne 8 < 0 x 2 A \ X 0 f 0 (x) = x 2 B \ X 0 : 2 else. Snce X 0 s dscrete, ths s automatcally contnuous. Now assume by nducton that we have f n : X n! [0, ] contnuous wth f n (A\X n ) 0 and f n (B \ X n ). Snce we have a pushout dagram ` Sn ` Dn X n X n, by the unversal property of the pushout, to defne f n on X n, we need only specfy a compatble par of functons on X n and on the dsjont unon. On X n, we take f n. To defne a map out of a D n, t s enough to defne a map on each D n. For each n-cell e,defnew D n closed by W n [ (A \ X n ) [ (B \ X n ). Defne g : W! [0, ] by 8 < f n ('(x)) x n g(x) = 0 x 2 : (A \ X n ) x 2 (B \ X n ). 69

3 We know that D n s compact Hausdor (or metrc) and thus normal. Thus, by the Tetze extneon theorem (32.) there s a Urysohn functon for the dsjont closed sets (A\X n ) and (B\X n ) whose restrcton n agrees wth f n '. Puttng all of ths together gves a Urysohn functon on X n for the A \ X n and B \ X n. By nducton, we are done. Even better, 44. Fr, Dec. 2 Theorem 44. (Lee, Theorem 5.22). Every CW complex s paracompact. Proposton Any CW complex X s locally path-connected. Proof. Let x 2 X and let U be any open neghborhood of x. We want to fnd a path-connected neghborhood V of x n U. Recall that a subset V X s open f and only f V \ X n s open for all n. WewlldefneV by specfyng open subsets V n X n wth V n+ \ X n = V n and then settng V = [V n. Suppose that x s contaned n the cell e n.wesetvk = ; for k<n.wespecfyv n by defnng j (V n ) for each n-cell e n j. If j 6=, weset j (V n )=;. We defne (V n ) to be an open n-dsc around (x) whose closure s contaned n (U). Now suppose we have defned V k for some k n. Agan, we defne V k+ by defnng each (V k+ ). By assumpton, (V k k+ (U). By the Tube lemma, there s an >0such that (usng radal coordnates) (V k ) (, ] U. Wedefne (V k+ )= (V k ) [, 2), whch s path-connected by nducton. Ths also guarantees that V k+ U \ X k+, allowng the nducton to proceed. Proposton 44.3 (Hatcher, A.). Any compact subset K of a CW complex X meets fntely many cells. Proof. For each cell e meetng K, pck a pont k 2 K \ e. Let S = {k }. We use proprety W to show that S s closed n X. It s clear that S \ X 0 s closed n X 0 snce X 0 s dscrete. Assume that S \ X n s closed n X n. Now n X n,thesets\ X n s the unon of the closed subset S \ X n and the ponts k that le n open n-cells. By Lemma 43., ths set of k s closed as well. The argument above n fact shows that any subset of S s closed, so that S s dscrete. But S s closed n K, sos s compact. Snce S s both dscrete and compact, t must be fnte. Corollary Any CW complex has the closure-fnte property, meanng that the closure of any cell meets fntely many cells. Proof. The closure of e s (D n ), whch s compact. The result follows from the proposton. Corollary () A CW complex X s compact f and only f t has fntely many cells. () A CW complex X s locally compact f and only f the collecton E of cells s locally fnte. We have talked recently about two good famles of spaces, CW complexes and manfolds. How are they related? A CW complex s a much more general knd of space. For nstance, S _ S has a perfectly good CW structure wth a sngle 0-cell and two -cells, but t s not a manfold snce the basepont does not have a Eucldean neghborhood. On the other hand, most manfolds do admt CW decompostons. Theorem 44.6 (Lee, 5.25). Every -manfold admts a (nce) CW decomposton. 70

4 Theorem 44.7 (Lee, 5.36, 5.37). Every n-manfold admts a (nce) CW decomposton for n =2, 3. Accordng to p. 529 of Allen Hatcher s Algebrac Topology book, t s an open queston whether or not every 4-manfold admts a CW decomposton. But n-manfolds for n 5 do always admt a CW decomposton. Another mportant problem, back purely n the realm of manfolds, s to try to lst all manfolds of a gven dmenson. Theorem 44.8 (Classfcaton of -manfolds). Every nonempty, connected -manfold M s homeomorphc to S f t s compact and to R f t s noncompact. For ths theorem, t wll be convenent to work wth nce CW structures. Defnton If X s a space wth a CW structure, we say that an n-cell e n s regular f the characterstc map : D n! e X s a homeomorphsm onto ts mage. We say that a CW complex s regular f every cell s regular. Proof. The frst step s to show that every -manfold has a regular CW decomposton. The man dea s to cover M by a countable collecton {U n } of regular charts (each closure U n n M should be homeomorphc to [0, ]). Then, usng nducton, t s possble to put a regular CW structure on U n = S n k= U k n such a way that U n U n+ s the ncluson of a subcomplex. (See Lee 5.25 for more detals.) Clearly, each -cell bounds two 0-cells, snce the -cell s assumed to be regular. Somewhat less clear s the fact that each 0-cell s n the boundary of two -cells (see Lee 5.26). We enumerate the 0-cells (aka vertces) and -cells (aka edges) n the followng way. Frst, pck some 0-cell, and call t v 0. Pck an edge endng at v 0, and call ths e 0. The other endpont of e 0 we call v. The other edge endng at v s called e. We can contnue n ths way to get v 2,v 3,... and e 2,e 3,... Now there s also another edge endng at v 0, whch should be called e. Let v be the other endpont. We can contnue to get v 2,v 3,... and e 2,e 3,... There are two cases to consder: Case : The vertces v, 2 Z are all dstnct. Then for each n 2 Z, we have an embeddng [n, n + ] n = [, ]! e n. These glue together to gve a contnuous map f : R! X. Our assumpton means that f s njectve when restrcted to Z. We can then see t s globally njectve snce ts restrcton to any (n, n + ) s a characterstc map for a cell (thus njectve) and all cells are dsjont. Next, we show that f s open. Any open subset of (n, n + ) s taken by f to an open subset of M, snce the top-dmenson cells are always open n a CW complex. It remans to show that f takes ntervals of the form (n, n + ) to open subsets of M. By takng small enough, we can ensure that ths mage s contaned n (the closure of) two -cells. We can then see that ths subset of M s open by pullng back along the characterstc maps (pullng back along these two characterstc maps wll gve half-open ntervals n D ). Snce M s connected, n order to show that f s surjectve, t now su ces to show that f(r) s closed. Let x2 f(r). If x les n a -cell e, thene s a neghborhood of x dsjont from f(r). The other possblty s that x s a 0-cell. But then x must be the endpont of two closed -cells e and e 0. Nether e nor e 0 can be n f(r) snce ths would mply that x also les n f(r). But then e [{x}[e 0 s a neghborhood of x dsjont from f(r). We have shown that f : R! M s an open, contnuous bjecton. So t s a homeomorphsm. Case 2: For some n 2 Z and k>0, we have v n = v n+k. We may then pck n and k so that k s mnmal. Then the vertces v n,...,v n+k are dstnct, as are the edges e n,...,e n+k. Ths mples that the restrcton of f to [n, n + k) s njectve. If we consder the restrcton only to the closed nterval [n, n + k], then we get a closed map, snce the doman s compact and the target s 7

5 Hausdor. We clam also that f([n, n+k]) s open n M. Indeed, f we pck any x 2 [n, n+k] whch les n an nterval (, + ), then the open -cell e s a neghborhood of f(x) that s contaned n the mage of f. If we consder any nteror nteger n<<n+ k, thene [{v }[e s an open neghborhood n the mage of f. Fnally, e n [{n}e n+k s a neghborhood of f(n) =f(n + k) n M. Snce the mage f([n, n + k]) s both closed and open n M and M s connected, we conclude that f([n, n + k]) = M. Sncef(n) =f(n + k), we get an nduced map f :[n, n + k] = S! M whch s a bjecton. Snce S s compact and M s Hausdor, ths s a homeomorphsm. We prevously also brefly mentoned the dea of a manfold wth boundary. There s a smlar result: Theorem Every nonempty, connected -manfold wth boundary s homeomorphc to [0, ] f t s compact and to [0, ) f t s noncompact. Next semester, we wll smlarly classfy all compact 2-manfolds (the lst of answers wll be a lttle longer). A closely related dea to CW complex s the noton of smplcal complex. A smplcal complex s bult out of smplces. By defnton, an n-smplex s the convex hull of n + a nely ndependent ponts n R k, for k n +. Ths means that after translatng ths set so that one pont moves to the orgn, the resultng collecton of ponts s lnearly ndependent. There s a standard n-smplex n R n+ defned by n = {(t 0,...,t n ) 2 R n+ X t =,t 0}. In general, f s an n-smplex generated by {t 0,...t n }, then the convex hull of any subset s called a face of the smplex. A (Eucldean) smplcal complex s then a subset of R k that s a unon of smplces such that any two overlappng smplces meet n a face of each. We also usually requre the collecton of smplces to be locally fnte. Snce an n-smplex s homeomorphc to D n, t can be seen that a smplcal complex s a regular CW complex. A decomposton of a manfold as a smplcal complex s known as a trangulaton of the manfold. Just as one can ask about CW structures on manfolds, one can also ask about trangulatons for manfolds. Theorem 44.. () Every -manfold s trangulable (ndeed, we know the complete lst of connected -manfolds). (2) Tbor Radó proved n 925 that every 2-manfold s trangulable. (3) Edwn Mose proved n the 950s that every 3-manfold s trangulable. (4) Mchael Freedman dscovered the 4-dmensonal E 8 -manfold n 982, whch s not trangulable. (5) Cpran Manolescu showed n March 203 that manfolds n dmenson 5 are not trangulable. 72