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2 THE HOMOTOPY GROUPS RELATED TO L 2 T (m)/(v 1 ) Zihong Yuan joint with Xiangjun Wang and Xiugui Liu School of Mathematical Sciences Nankai University Dec.16, 2008 The Second East Asia Conference on Algebraic Topology Page 2 of 20

3 The homotopy groups related to L 2 T (m)/(v 1 ) Part I Part II Toward the homotopy group of L 2 T (m)/(v 1 ) Part III Construction of the cocycle for m > 1 Page 3 of 20

4 1 We will work in the stable homotopy category, all the spectra(spaces) will be localized at an odd prime p. The generalized homologies: In the stable homotopy category, the generalized homology is referred as a spectrum. (1) BP-spectrum : BP =π (BP ) = Z (p) [v 1, v 2,, v n, ] Z (p) ={ m p n} n Page 4 of 20

5 BP (BP ) = π (BP BP ) = BP [t 1, t 2,, t n, ] (2) E(n): Consider the BP -modules E(n) = vn 1 BP /(v n+i i > 0) = Z (p) [v 1,, v n 1, v n, vn 1 ] E(0) is the rational Q. By the exact functor theorem of Landweber, that E(n) (X) = E(n) BP BP (X) is a homology theory. Then using the representation theorem, we have a spectrum E(n) such that E(n) (X) = π (E(n) X) Page 5 of 20

6 (3) E m (2): Consider the E(2) -modules, E m (2) = E(2) [v 3, v m+2 ] = Z (p) [v 1, v ±1 2, v 3,, v m+2 ] E m (2) is also a spectrum. Localization with respect to generalized homology If E is a spectrum, a spectrum X is E-acyclic if E X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X Y is an E-equivalence if it induces an isomorphism on E -homologies. Bousfield shows that there is a functor called E-localization L E X, and a natural transformation X L E X which is an E-equivalence. Page 6 of 20

7 Examples: A spectrum X localized with respect to the Eilenberg-MacLane spectrum KZ/p is the spectrum itself, i.e., L KZ/p X = X. We also have L BP X = X. The localization with respect to the Johnson-Wilson spectrum E(n) is referred to L n, i.e., L n X = L E(n) X. By the chromatic convergence theorem, there is a tower L n X L n 1 X L 0 X (1.1) for any finite spectrum X. And the inverse limit of L n X is X. So from L E X, we could study the part of π (X) which E sees. Page 7 of 20

8 The Ravenel spectrum T(m) and homotopy groups of localized spectra: The Ravenel spectrum T (m) is characterized by BP -homology BP (T (m)) = BP [t 1,, t m ] BP (BP ) = BP [t 1, t 2,, t n, ]. There is a tower of spectra S 0 = T (0) T (1) T (2) T (n) BP π (L 2 S 0 ) Page 8 of 20 π (L 2 S 0 ) for p > 3 is detected by Shimomura and Yabe. ( Topology, 1995)

9 Shimomura and Wang determined π (L 2 S 0 ) at the prime 3. (Topology, 2002) Shimomura and Wang determined the Adams-Novikov E 2 -term for π (L 2 S 0 ) at the prime 2 also. But because of the Adams-Novikov differentials, it is far from the full understanding of the homotopy groups. π (L 2 T (m)): Let T (m)/(v 1 ) denote the cofiber of the self map v 1 : Σ 2(p 1) T (m) T (m). Then by the cofiber sequence: Σ 2(p 1) T (m) v 1 T (m) T (m)/(v1 ) We should consider L 2 T (m)/(v 1 ). In this lecture,we will consider the homotopy groups of L 2 T (m)/(v 1 ). Page 9 of 20

10 2 Toward the homotopy group of L 2 T (m)/(v 1 ) For a spectrum X, a basic method to compute its hopmotopy groups is the Adams-Novikov spectral sequence, the E 2 -term of the spectral sequence is Ext s,t, BP (BP ) (BP, BP (X)) and it converges to π (X). To compute the E 2 -term Ext s,t, BP (BP ) (BP, BP (X)), we have the chromatic spectral sequence and the Bockstein spectral sequence. Page 10 of 20

11 The chromatic spectral sequence: Let T (m)/(p, v 1 ) be the cofiber of the localization map T (m)/(v 1 ) L 0 T (m)/(v 1 ) T (m) L 0 T (m) T (m)/(p ) Apply the localization functor L 2, we have the following cofiber sequence: L 2 T (m)/(v 1 ) L 0 T (m)/(v 1 ) L 2 T (m)/(p, v 1 ) (2.1) These cofiber sequences induce long exact sequences of homotopy groups. Thus it suffices to compute the homotopy groups π (L 0 T (m)/(v 1 )), π (L 2 T (m)/(p, v 1 )) The homotopy groups of π (L 0 T (m)/(v 1 )) is known to be rational homotopy. Page 11 of 20

12 The Bockstein spectral sequence: To compute π (L 2 T (m)/(p, v 1 )), denote the cofiber of T (m) T (m)/(p) p T (m) T (m) T (m)/(p). There is the cofiber sequence L 2 T (m)/(p, v 1 ) L 2 T (m)/(p, v 1 ) which induces a long exact sequence of homotopy groups. p T (m) by p L 2 (T (m)/(p, v 1 ) (2.2) Page 12 of 20

13 Let Γ(2, m) = E m (2) [t m+1, t m+2, ] BP E m (2) By change of rings theorem, we could get Ext s,t, BP BP (BP, BP (L 2 (T (m) X)) = Ext s,t Γ(2,m) (E m(2), E m (2) (X)). By the equations above, we simplify all the items from (BP, BP BP ) to (E m (2), Γ(2, m)). Let M2 0 = E m (2) /(p, v 1 ),L 1 1 = E m (2) /(p, v 1 ),L(k, 1) = E m (2) /(p k, v 1 ), Ext s,t Γ(2,m) (E m(2), M) = H s,t (M) for short. Then we will have the short exact sequence of E m (2) E m (2)-comodules 0 M 0 2 L 1 1 p L Page 13 of 20

14 This short exact sequence induces the following long exact sequence H s, M 0 2 H s, L 1 1 p H s, L 1 1 δ H s+1, M 0 2. H, M 0 2 = Z/p[v ±1 2, v 3,, v m+2 ] E[h 0 m+1, h 1 m+1, h 0 m+2, h 1 m+2] where h j i denotes the cohomology class represented by t pj i. Page 14 of 20

15 The connecting homomorphism δ: A simple description of δ is necessary. By the long exact sequence above and the structure of H, M 0 2, the essential part is the differential of t p m+1 and tp m+2. A cocycle of t p m+1 or tp m+2 is necessary, if not, the structure of δ will be too complex to compute. When m = 1, we could find a cocycle in H, L 1 1, and finally we get π (L 2 T (1)/(v 1 )). while m > 1,unluckily, we did not find a cocycle in H, L 1 1. Page 15 of 20

16 L 2 T(m)/(p [ m+1 2 ]+2, v 1 ): We find cocycle just in H s, L(k, 1) while k = [ m+1 2 ] + 1. Finally, we get the homotopy group of L 2 T (m)/(p [ m+1 2 ]+2, v 1 ). When m > 1,the short exact sequence will be 0 M 0 2 p k L(k + 1, 1) L(k, 1) 0. This short exact sequence induces the following long exact sequence Page 16 of 20 H s, M 0 2 p k H s, L(k + 1, 1) H s, L(k, 1) δ H s+1, M 0 2.

17 π (L 2 T(1)/(v 1 )): C 0 =Z (p) {v spn 2 v tpm 3 /p min{m,n}+1 0 m, n }, C 1 = C 0 1 C 1 1, C 1 0 = Z (p) {v spn 2 v tpm 1 3 h 0 3/p n+1 0 n < m < }, C 1 1 = Z (p) {v spn 1 2 v tpm 3 h 0 2/p m+1 0 m n }, h 1 3 C 0 =Z (p) {v spn 2 v tpm 3 h 1 3/p}. Theorem 2.3 H, L 1 1 and then the homotopy groups of L 2 T (1)/(p, v 1 ) are isomorphic to the direct sum of ( C 0 C 1 ) E[h 1 2] and h 1 3 C 0 E[h 0 2, h 0 3]. Page 17 of 20 Theorem 2.4 The homotopy groups of L 2 T (1)/(v 1 ) are isomorphic to π (L 2 T (1)/(v 1 )) = Z (p) (π (L 2 T (1)/(p, v 1 )) Q/Z (p) ).

18 3 Construction of the cocycle for m > 1 We do it by the structure map of (E m (2), Γ(2, m)): In (BP, BP BP ),we have (t n ) = t i t pi j i+j=n n 1 pb k n = i=1 + v n 1b n v n 2 b n v 1 b 0 n 1 v pk+1 i b k+i n i + t pk+1 i i+j=n t pi+k+1 j (t n ) pk+1. Page 18 of 20

19 Then, in (E m (2), Γ(2, m)), we compute until n = 2m + 3. After that, we construct elements c i,j (m + 1 i 2m + 1) such that d(c i,j ) = p [ i+1 2 ] b j 1 m+i With these elements, we can eliminate the items b j i. Finally we will get a cocycle. Page 19 of 20

20 THANK YOU VERY MUCH!!! Page 20 of 20