Contents 1 The homotopy elements h 0 h n 2 Toda differential 3 Method - PowerPoint PPT Presentation

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration On the homotopy elements h 0 h n Xiangjun Wang SUSTech School of Mathematical Sciences, Nankai University June 6, 2018 Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Contents 1 The homotopy elements h 0 h n 2 Toda differential 3 Method of infinite descent 4 Further consideration Xiangjun Wang On the homotopy elements h 0 h n

� � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Classical ASS and ANSS • Let p � 5 be an odd prime. One has the classical Adams spectral sequence (ASS) and the Adams-Novikov spectral sequence (ANSS), they all converge to the stable homotopy groups of spheres. E 2 = Ext s,t { E s,t ⇒ π ∗ ( S 0 r , d r } = p ) BP ∗ BP ( BP ∗ , BP ∗ ) Φ Φ E 2 = Ext s,t { E s,t ⇒ π ∗ ( S 0 r , d r } = p ) A ( Z /p, Z /p ) Between the ANSS and the ASS there is the Thom map Φ induced by Φ : BP − → H Z /p. Xiangjun Wang On the homotopy elements h 0 h n

� � � � � � � � � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration To detect the E 2 -terms of the ASS and of the ANSS, one has the following spectral sequences MSS MSS BSS � CSS � H ∗ ( P, Q ) CESS = � Ext s,t H ∗ ( q − 1 H ∗ ( q − 1 Q/ ( q ∞ · · · q ∞ Q/ ( q 0 · · · q n − 1 )) n − 1 ) n n 0 A ∼ Alg. NSS Alg. NSS Alg. NSS ASS Φ n − 1 )) CSS ANSS BP ∗ / ( p ∞ · · · v ∞ � Ext s,t H ∗ ( v − 1 � H ∗ ( v − 1 π ∗ ( S 0 BP ∗ / ( p · · · v n − 1 )) p ) n n BP ∗ BP BSS MSS where P = Z /p [ ξ 1 , ξ 2 , · · · ] and Q = Z /p [ q 0 , q 1 , · · · ] . Xiangjun Wang On the homotopy elements h 0 h n

� � � � � � � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration The homotopy elements h 0 h n • One has β p n /p n − 1 ∈ Ext 2 , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) , which is detected by the CSS and Φ( β p n /p n − 1 ) = h 0 h n +1 . H ∗ ( P, Q ) CESS Ext 2 A ∼ = Alg. NSS ASS Φ � Ext 2 � π ∗ ( S 0 ) H 0 ( v − 1 2 BP ∗ / ( p ∞ , v ∞ 1 )) CSS BP ∗ BP ANSS Ext 2 , ∗ π ∗ ( BP ∧ � β p n /p n − 1 ∈ BP ∗ BP ( BP ∗ , BP ∗ ) ⊂ X 2 ) Φ Ext 2 , ∗ h 0 h n +1 ∈ A ( Z /p, Z /p ) ⊂ π ∗ ( H ∧ X 2 ) Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • The convergence of h 0 h n +1 in the classical ASS (that of β p n /p n − 1 in the ANSS) have been being a long standing problem in stable homotopy groups of spheres. • Let M be the mod p Moore spectrum, M (1 , p n − 1) be the cofiber of v p n − 1 : Σ ∗ M − → M . 1 Secondary periodic family elements in the ANSS, D. Ravenel Theorem Let p � 5 be an odd prime. If for some fixed n � 1 , • the spectrum M (1 , p n − 1) is a ring spectrum, • β p n /p n − 1 is a permanent cycle and • the corresponding homotopy element has order p , then β sp n /j is a permanent cycle (and the corresponding homotopy element has order p ) for all s � 1 and 1 � j � p n − 1 . Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • The convergence of h 0 h n +1 in the classical ASS (that of β p n /p n − 1 in the ANSS) have been being a long standing problem in stable homotopy groups of spheres. • Let M be the mod p Moore spectrum, M (1 , p n − 1) be the cofiber of v p n − 1 : Σ ∗ M − → M . 1 Secondary periodic family elements in the ANSS, D. Ravenel Theorem Let p � 5 be an odd prime. If for some fixed n � 1 , • the spectrum M (1 , p n − 1) is a ring spectrum, • β p n /p n − 1 is a permanent cycle and • the corresponding homotopy element has order p , then β sp n /j is a permanent cycle (and the corresponding homotopy element has order p ) for all s � 1 and 1 � j � p n − 1 . Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • The convergence of h 0 h n +1 in the classical ASS (that of β p n /p n − 1 in the ANSS) have been being a long standing problem in stable homotopy groups of spheres. • Let M be the mod p Moore spectrum, M (1 , p n − 1) be the cofiber of v p n − 1 : Σ ∗ M − → M . 1 Secondary periodic family elements in the ANSS, D. Ravenel Theorem Let p � 5 be an odd prime. If for some fixed n � 1 , • the spectrum M (1 , p n − 1) is a ring spectrum, • β p n /p n − 1 is a permanent cycle and • the corresponding homotopy element has order p , then β sp n /j is a permanent cycle (and the corresponding homotopy element has order p ) for all s � 1 and 1 � j � p n − 1 . Xiangjun Wang On the homotopy elements h 0 h n

� � � � � � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • S. Oka proved that M (1 , p n − 1) is a ring spectrum. • From the theorem above and the convergence of h 0 h n +1 one can prove the β p n /p n − 1 is a permanent cycle of order p . • S ∗ � 0 β pn/pn − 1 β pn/pn − 1 p � S 0 � S 0 Σ − 1 M People concerned with the triviality of v p n − 1 � β p n /p n − 1 1 S ∗ v pn 0 � 2 � β pn/pn − 1 v pn − 1 Σ ∗ M (1 , p n − 1) � Σ − 1 M � Σ ∗ M 1 Xiangjun Wang On the homotopy elements h 0 h n

� � � � � � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • S. Oka proved that M (1 , p n − 1) is a ring spectrum. • From the theorem above and the convergence of h 0 h n +1 one can prove the β p n /p n − 1 is a permanent cycle of order p . • S ∗ � 0 β pn/pn − 1 β pn/pn − 1 p � S 0 � S 0 Σ − 1 M People concerned with the triviality of v p n − 1 � β p n /p n − 1 1 S ∗ v pn 0 � 2 � β pn/pn − 1 v pn − 1 Σ ∗ M (1 , p n − 1) � Σ − 1 M � Σ ∗ M 1 Xiangjun Wang On the homotopy elements h 0 h n

� � � � � � The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration • S. Oka proved that M (1 , p n − 1) is a ring spectrum. • From the theorem above and the convergence of h 0 h n +1 one can prove the β p n /p n − 1 is a permanent cycle of order p . • S ∗ � 0 β pn/pn − 1 β pn/pn − 1 p � S 0 � S 0 Σ − 1 M People concerned with the triviality of v p n − 1 � β p n /p n − 1 1 S ∗ v pn 0 � 2 � β pn/pn − 1 v pn − 1 Σ ∗ M (1 , p n − 1) � Σ − 1 M � Σ ∗ M 1 Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Toda differential • α 1 and b 0 = β 1 in Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) are permanent cycles in the ANSS, they converges to the homotopy elements α 1 , β 1 respectively. • H. Toda proved that α 1 β p 1 = 0 in π ∗ ( S 0 ) . • The relation α 1 β p 1 = 0 support a Adams differential d r ( x ) = α 1 b p 0 . It is detected that x = b 1 i.e d 2 p − 1 ( b 1 )) = k · α 1 b p 0 • Based on d 2 p − 1 ( b 1 ) = k · α 1 b p 0 , D. Ravenel proved that d 2 p − 1 ( b n ) ≡ α 1 b p n − 1 Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Toda differential • α 1 and b 0 = β 1 in Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) are permanent cycles in the ANSS, they converges to the homotopy elements α 1 , β 1 respectively. • H. Toda proved that α 1 β p 1 = 0 in π ∗ ( S 0 ) . • The relation α 1 β p 1 = 0 support a Adams differential d r ( x ) = α 1 b p 0 . It is detected that x = b 1 i.e d 2 p − 1 ( b 1 )) = k · α 1 b p 0 • Based on d 2 p − 1 ( b 1 ) = k · α 1 b p 0 , D. Ravenel proved that d 2 p − 1 ( b n ) ≡ α 1 b p n − 1 Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Toda differential • α 1 and b 0 = β 1 in Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) are permanent cycles in the ANSS, they converges to the homotopy elements α 1 , β 1 respectively. • H. Toda proved that α 1 β p 1 = 0 in π ∗ ( S 0 ) . • The relation α 1 β p 1 = 0 support a Adams differential d r ( x ) = α 1 b p 0 . It is detected that x = b 1 i.e d 2 p − 1 ( b 1 )) = k · α 1 b p 0 • Based on d 2 p − 1 ( b 1 ) = k · α 1 b p 0 , D. Ravenel proved that d 2 p − 1 ( b n ) ≡ α 1 b p n − 1 Xiangjun Wang On the homotopy elements h 0 h n

The homotopy elements h 0 h n Toda differential Method of infinite descent Further consideration Toda differential • α 1 and b 0 = β 1 in Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) are permanent cycles in the ANSS, they converges to the homotopy elements α 1 , β 1 respectively. • H. Toda proved that α 1 β p 1 = 0 in π ∗ ( S 0 ) . • The relation α 1 β p 1 = 0 support a Adams differential d r ( x ) = α 1 b p 0 . It is detected that x = b 1 i.e d 2 p − 1 ( b 1 )) = k · α 1 b p 0 • Based on d 2 p − 1 ( b 1 ) = k · α 1 b p 0 , D. Ravenel proved that d 2 p − 1 ( b n ) ≡ α 1 b p n − 1 Xiangjun Wang On the homotopy elements h 0 h n