The differential in the Adams spectral sequence for spheres

• Main Authors: Chung-Wei Kan, 甘崇瑋
• Other Authors: Wen-Hsiung Lin
• Format: Others
• Language: en_US
• Published: 2001
• Online Access: http://ndltd.ncl.edu.tw/handle/64596022888616182984

博士 === 國立清華大學 === 數學系 === 89 === The differential in the Adams spectral sequence for spheres Let A denote the mod 2 Steenrod algebra . The mod 2 Adams spectral sequence is one of the most important tools for computing the 2-adic stable homotopy groups of spheres , which has E_2 term =Ext group , the cohomology of the mod 2 Steenrod algebra . Let h{i} be the class corresponding to the generator sq{2^{i}}in A as described by J. F. Adams in [ 1] . Adams also proves that h{i}^{2} in Ext group and that h{i}^{3}=h{i-1}^{2}h{i+1} in Ext group for all i>=0 . It is well known that h{i}^{3} , for 0<=i <=3 , detect homotopy classes . Mahowald and Tangora have shown in [ 13] that h{4}^{3} survives in the Adams spectral sequence for spheres . W. H. Lin describes a method in [ 13] to suggest that h{i}^{3}) not survives for i>=6 Because of the difficulties of the calculations involved , W. H. Lin only give a complete proof of the case i=7 in [ 11] . The mail result of this thesis is h{5}^{3} not to detect homotopy classes in the Adams spectral sequence for spheres . This thesis is organized as follows . All spaces and maps to be considered are stable objects with base points . For a homotopy element, its representative is also denoted by the same notation if there are no ambiguity. All homology and cohomology groups have the mod 2 coefficients . To show ( 1.2) ( 1.3) we need some preliminaries on the cohomology of the mod 2 Steenrod algebra . These Ext groups will be calculated by the May spectral sequence which is recalled in Section 2 . In Section 3 we will show another tool , the lambda algebra Lambda ( [ 6] ) for computing some Ext groups on spheres , projective spaces and stunted projective spaces . In Section 4 we describe some well known properties about the differentials in the Adams spectral sequence . In Section 5 we prove ( 1.2) and ( 1.3) .