Localization of epimorphisms and monomorphisms in Homotopy theory II

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1 Localizatio o eimorhisms ad moomorhisms i Homotoy theory II She Wehuai Deartmet o Mathematics, South hia Normal Uiversity, Guagzhou, 5063, P.R. o hia Abstract: I this ote, we rove that localizatio o saces reserves eimorhisms ad moomorhisms i homotoy theory i some coditios. Keywords: Localizatio;Homotoy-eimorhism;Homotoy-moomorhism MR 2000 lassiicatio: 55P60. Itroductio Recall that : HW, the homotoy category o oited ath-coected W-saces, is a homotoy eimorhism (moomorhism) i give u,v : Z HW (u,v : Z HW ), uo = vo imlies u = v (o u = o v imlies u = v). I [4], we roved the ollowig theorem, aswerig a guestio osed by Hilto ad Roitberg i [3] whether localizatio o ilotet saces reserves homotoy eimorhisms ad homotoy moomorhisms. Theorem. I : is a homotoy eimorhism (moomorhism) o ilotet saces, the the -localized ma : is also a homotoy eimorhism (moomorhism). Here is a rime or 0. I this ote, we shall show that Theorem is true or some o-ilotet saces. It is iterestig ad diicult to study whether localizatio o saces i asacuberta-peschke s sece reserves eimorhisms ad moomorhisms i homotoy theory. The mai result o the ote is the ollowigs. Theorem 2. Let : be a homotoy eimorhism such that () π () is a isomorhism. I it satisies oe o the ollowig coditios, () π () is a iite grou, (2) π () is a ilotet torsio grou, (3) π () is a - torsio grou, the the -localized ma : is also a homotoy eimorhism. Theorem 3. Let : be a homotoy moomorhism such that () π () is a isomorhism. I π () is a - torsio grou, the the -localized ma : is also a homotoy moomorhism. -mail address: she@scu.edu.c

2 2. Lemmas Lemma [4]. () Let be a homotoy ushout. The : is a homotoy eimorhism i ad oly i j =j 2. (2) Let j 2 i 2 i 2.2 be a homotoy ullback. The : is a homotoy moomorhism i ad oly i i =i 2. Lemma 2. Let : be a ma such that () π () is a isomorhism. I ad are -coected, the the -localizatio o the square (2.) j 2. is also a homotoy ushout. j 2 j 2.3 Proo. Recall that i l : W W is the -localizatio o sace W, the l : H ) H ) is a -equivalet, ad l ) π ) is the -localizatio o π ) (see[]). I W is -coected, the H ) is -local or all (see[]). Hece l : H ) H ) is the -localizatio o H ), i.e., H ) H ) Z, where Z is the -localizatio o iterger abelia grou. Suose : is a ma such that () π () is a isomorhism. sice () π () is a isomorhism, ad the square (2.) is a homotoy ushout, the j = j 2 () π () is also a isomorhism. This imlies that j * = j 2* : π ( ) π ( ) is also a isomorhism. Thus, i ad are -coected, the is also -coected. Let be a homotoy ushout. The / is -coected, ad there is a maϕ : ' yieldig a homotoy commutative diagram j' 2 / j / 2.4

3 j / j / j 2 ad hece a ma o the Mayer-Vietoris sequece o the square (2.4) to the -localizatio o Mayer-Vietoris sequece o the square (2.), Sice H () Z H ( ), H () Z H ( ), H () Z H ( ). I this ma o Mayer-Vietor sequeces all grous excet H ( ' ) are maed by the idetity. This ϕ iduces a isomorhism o homology grous. Sice / ad is -coected, this ollows that ϕ is a homotoy equivalece. The the square (2.3) is a homotoy ushout. Lemma 3[4]. I ad are ilotet, the the -localizatio o the square (2.2) ϕ is also a homotoy ullback. i i2 Lemma 4[2]. Let F K(G,) be the ibratio sequee. I G is a -torsio grou, the F K (G,) is also a ibratio sequece. Lemma 5. Let F B ϕ F / / B / be homotoy commutative diagram o ibratio sequeces. I ϕ is a homotoy equivalece, the its let square is a homotoy ullback. Proo. osider the ollowig homotoy commutative diagram F F / / * B ϕ * B / where the rot ad back aces are homotoy ullbacks. I φ is a homotoy equivalece, the the bottom ace is homotoy ullback. This ollows that the to ace is a homotoy ullback.

4 Lemma 6[5]. Suose that we have a homotoy commutative cube A / B / / A D / B D i which all vertical aces are homotoy ullbacks. I oe o the to or bottom aces is a homotoy ushout, the so is the other. 3. Proos Theorem 4. Let : be a homotoy eimorhism ad () π () be a isomorhism. I there is a ormal subgrou N o π (), such that π ()/N is a -torsio grou ad N is trivial, the the -localized ma : is also a homotoy eimorhism. Proo. Let : be a homotoy eimorhisms such that () π () is a isomorhism. Deote ad are the coverig saces o,, resectively, or the subgrou N o π () π (). The we have the ollowig homotoy commutative diagram o coverig ibratio sequeces: K(π ()/N,) * K(π ()/N,) 2.5. I π ()/N π ()/N is the -torsio grou, by Lemma 4 ad the diagram (2.5), the the ollowig the ibratio sequeces diagram is homotoy commutative: Let K(π ()/N,) * K(π ()/N,) 2.6. be a homotoy ushout. The j =j 2 by Lemma (). Note that j * =j 2* () π () is a isomorhism. Deote j 2 j is the coverig sace o or subgrou N π ().

5 osider the ollowig homotoy commutative cube: 2.7 i which all vertical aces are homotoy ullbacks by Lemma 5 ad the diagram (2.5). Hece, the to ace is a homotoy ushout by Lemma 6. osider the -localizatio o the diagram (2.7), that is, the ollowig homotoy commutative cube: 2.8 j i which all vertical aces are homotoy ullbacks by Lemma 5 ad the diagram (2.6). Sice π ( ) π ( )= N, so π ( ) π ( )= N, but N is trivial, j 2 the ad is -coected. By Lemma 2, the to ace o the diagram (2.8) is a homotoy ushout. This ollows rom Lemma 6 that the bottom ace o the diagram (2.8) is a homotoy ushout. Sice j =j 2 imlies j =j 2, the : is a homotoy eimorhism by Lemma (). Proo o Theorem 2. Obvious, i π () satisies oe o the coditios o Theorem 2, the there must be a ormal subgrou N o π (), such that π ()/N is a -torsio grou ad N is trivial. Hece, Theorem 4 ollows rom Theorem 2. Proo o Theorem 3. Let : be a homotoy moomorhism such that : π () π () is a isomorhism. Suose π () is a - torsio grou. I the roo o Theorem 4, let N = 0, ad the ad are the uiversal coverig saces o,, resectively. Hece, we have the ollowig homotoy commutative diagram o ibratio sequeces: K( π (),) * K( π (),) 2.9. Note that π () is a - torsio grou, by lemma 4, the ollowig the ibratio

6 sequeces diagram is homotoy commutative: Let K( π (),) * K( π (),) i 2 i 2.0. be a homotoy ullback. The i =i 2 by Lemma (2). Deote H = Ker {i * =i 2* () π ()}, ad is the coverig sace o or the subgrou H o π (). Thus we have the ollowig homotoy commutative cube, i which all vertical aces o the diagram (2.) are homotoy ullbacks by Lemma 5 ad the diagram (2.9) 2.. osider the -localizatio o the diagram (2.), that is, the ollowig homotoy commutative cube: i2 2.2 i i which all vertical aces are homotoy ullbacks by Lemma 5 ad the diagram (2.0). Note that ad are -coected. By lemma 3, the to ace o the diagram (2.2) is a homotoy ullbck. This imlies that the bottom ace o the diagram (2.2) is a homotoy ullbck. Sice i =i 2 imlies i =i 2, the : is a homotoy moomorhism by Lemma (2).

7 Reereces [].asacubarta ad G.Peschke, Localizig with resect to sel mas o the circle, Tras. Amer. Math. Soc., 339(993), [2].asacubarta, O Structures reserved by idemotet trasormatio o grou ad homotoy tyes, otemorary Math., 262(2000), [3] P.Hilto ad J.Roitberg, Relatio eimorhisms ad moomorhisms i homotoy theory, omositio Math., 6(987), [4] H.Li ad W.H.She, Localizatio o eimorhisms ad moomorhisms i homotoy theory, omositio Math., 9(994), [5] M.Mather, Pull-backs i homotoy theory, a.j.math., 28(976),