## The triviality of the 61-stem in the stable homotopy groups of spheres.(English)Zbl 1376.55013

Let $$\theta_n\in \pi_{2^{n+2}-2}$$ denote the Kervaire invariant one element where we know that $$\theta_1$$, $$\theta_2$$ and $$\theta_3$$ can be chosen to be $$\eta^2$$, $$\nu^2$$ and $$\sigma^2$$ respectively, but it does not exist for all $$n \geq 7$$. Moreover, it is known that both $$\theta_4$$ and $$\theta_5$$ exist as elements such that $$2\theta_4=2\theta_5=0$$ and $$\theta_4^2=\theta_5^2=0$$. The purpose of this paper is to prove the following: (i) The 2-primary component of $$\pi_{61}$$ is zero; (ii) $$\theta_5$$ is contained in the strictly defined 4-fold Toda bracket $$\langle 2, \theta_4, \theta_4, 2 \rangle$$; (iii) $$S^{61}$$ has a unique smooth structure. The last two assertions (ii) and (iii) are proved in the first section, assuming that (i) holds true, which is roughly sketched below. The proof of (i) is given in Section 2 or later. Since $$\langle 2, \theta_4, \theta_4 \rangle \subset {}_2\pi_{61}$$, we have $$\langle 2, \theta_4, \theta_4 \rangle=0$$ due to (i) so we see that $$\langle 2, \theta_4, \theta_4, 2 \rangle$$ is strictly defined. If $$h_n \in \text{Ext}_2^{1, 2^n}$$ is an element in the Adams spectral sequence (ASS) for spheres such that its square $$h_n^2$$ detects $$\theta_n$$, then because of $$d_2(h_5)=h_0h_4^2$$ it follows that $$\langle h_0, h_4, h_4, h_0 \rangle=h_5^2$$ holds in the $$E_3$$-page. From this (ii) follows immediately. Let $$\Theta_n$$ be the group of $$h$$-cobordism classes of homotopy $$n$$-spheres and $$\Theta_n^{bp}$$ the subgroup defined by homotopy spheres which bound parallelizable manifolds. Then from the fact that $$\theta_5$$ is a Kervaire invariant one element we have that $$\Theta_{61}^{bp}=0$$, so that $$\Theta_{61}=\pi_{61}/\text{Im} J$$. This implies that when shown that $$\text{Coker} J$$ in dimension 61 vanishes at all odd primes, we obtain $$\Theta_{61}=0$$ by combining this fact with (i) and thus we have (iii).
The proof of (i) essentially follows the computation of $$\pi_*$$ by D. C. Isaksen [“Classical and motivic Adams charts”, Preprint, arXiv:1401.4983], developing a new method which allows us to overcome the lack of computability of differentials in the ASS for the 2-local sphere. In fact, we have three elements in the $$E_\infty$$-page whose differentials are unknown. But by determining their respective differentials based on this new method, the authors succeed in proving that there are no elements left in the $$E_\infty$$-page of the 61-stem. In terms of the notation used therein, they are described as follows: $d_3(D_3)=B_3, \;d_5(A')=h_1B_{21} \;\text{and} \;d_4(h_1X_1)=gz.$ Here the last two relations are obtained, respectively, in Sections 12 and 11. However, the proof of the first relation requires a lengthy technical argument in Sections 3-10.

### MSC:

 55Q45 Stable homotopy of spheres
Full Text:

### References:

 [1] Adams, J. F., On the structure and applications of the {S}teenrod algebra, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 32, 180-214, (1958) · Zbl 0083.17802 [2] Adams, J. F., On the non-existence of elements of {H}opf invariant one, Ann. of Math. (2). Annals of Mathematics. Second Series, 72, 20-104, (1960) · Zbl 0096.17404 [3] Aubry, Marc, Calculs de groupes d’homotopie stables de la sph\“ere, par la suite spectrale d”{A}dams-{N}ovikov, Math. Z.. Mathematische Zeitschrift, 185, 45-91, (1984) · Zbl 0509.55009 [4] Bauer, Tilman, Computation of the homotopy of the spectrum {\tt tmf}. Groups, Homotopy and Configuration Spaces, Geom. Topol. Monogr., 13, 11-40, (2008) · Zbl 1147.55005 [5] Becker, J. C.; Gottlieb, D. H., The transfer map and fiber bundles, Topology. Topology. An International Journal of Mathematics, 14, 1-12, (1975) · Zbl 0306.55017 [6] Behrens, Mark; Hill, Mike; Hopkins, Mike; Mahowald, Mark, Exotic spheres detected by topological modular forms [7] Barratt, M. G.; Jones, J. D. S.; Mahowald, M. E., Relations amongst {T}oda brackets and the {K}ervaire invariant in dimension {$$62$$}, J. London Math. Soc. (2). Journal of the London Mathematical Society. Second Series, 30, 533-550, (1984) · Zbl 0606.55010 [8] Barratt, M. G.; Jones, J. D. S.; Mahowald, M. E., The {K}ervaire invariant problem. Proceedings of the {N}orthwestern {H}omotopy {T}heory {C}onference, Contemp. Math., 19, 9-22, (1983) · Zbl 0528.55010 [9] Barratt, M. G.; Mahowald, M. E.; Tangora, M. C., Some differentials in the {A}dams spectral sequence. {II}, Topology. Topology. An International Journal of Mathematics, 9, 309-316, (1970) · Zbl 0213.24901 [10] Bruner, Robert, A new differential in the {A}dams spectral sequence, Topology. Topology. An International Journal of Mathematics, 23, 271-276, (1984) · Zbl 0565.55024 [11] Bruner, Robert R., The cohomology of the mod 2 {S}teenrod algebra: a computer calculation [12] Cohen, Joel M., The decomposition of stable homotopy, Ann. of Math. (2). Annals of Mathematics. Second Series, 87, 305-320, (1968) · Zbl 0162.55102 [13] Connell, E. H., A topological {$$H$$}-cobordism theorem for {$$n\geq 5$$}, Illinois J. Math.. Illinois Journal of Mathematics, 11, 300-309, (1967) · Zbl 0146.45201 [14] Freedman, Michael Hartley, The topology of four-dimensional manifolds, J. Differential Geom.. Journal of Differential Geometry, 17, 357-453, (1982) · Zbl 0528.57011 [15] Freudenthal, H., \“{U}ber die Klassen der Sph\'”{a}renabbildungen. {I.G}ro{\ss}e Dimensionen, Comp. Math., 5, 299-314, (1938) · JFM 63.1161.02 [16] Henriques, Andre, The homotopy groups of $$tmf$$ and of its localizations. Topological {M}ocular {F}orms, Math. Surveys Monogr., 201, 189-205, (2014) · Zbl 1328.55015 [17] Hill, M. A.; Hopkins, M. J.; Ravenel, D. C., On the nonexistence of elements of {K}ervaire invariant one, Ann. of Math. (2). Annals of Mathematics. Second Series, 184, 1-262, (2016) · Zbl 1366.55007 [18] Hopf, Heinz, \“{U}ber die {A}bbildungen der dreidimensionalen {S}ph\'”are auf die {K}ugelfl\"ache, Math. Ann.. Mathematische Annalen, 104, 637-665, (1931) · Zbl 0001.40703 [19] Isaksen, Daniel C., Stable stems, (2014) · Zbl 1421.55011 [20] Isaksen, Daniel C., Classical and motivic {A}dams charts, (2014) [21] Isaksen, Daniel C.; Xu, Zhouli, Motivic stable homotopy and the stable 51 and 52 stems, Topology Appl., 190, 31-34, (2015) · Zbl 1327.55007 [22] Kervaire, Michel A.; Milnor, John W., Groups of homotopy spheres. {I}, Ann. of Math. (2). Annals of Mathematics. Second Series, 77, 504-537, (1963) · Zbl 0115.40505 [23] Kochman, Stanley O.; Mahowald, Mark E., On the computation of stable stems. The \v{C}ech Centennial, Contemp. Math., 181, 299-316, (1995) · Zbl 0819.55006 [24] Kochman, Stanley O., Stable Homotopy Groups of Spheres. A computer-assisted approach, Lecture Notes in Math., 1423, viii+330 pp., (1990) · Zbl 0688.55010 [25] Kahn, Daniel S.; Priddy, Stewart B., The transfer and stable homotopy theory, Math. Proc. Cambridge Philos. Soc.. Mathematical Proceedings of the Cambridge Philosophical Society, 83, 103-111, (1978) · Zbl 0373.55011 [26] Lin, W\^en Hsiung, Algebraic {K}ahn-{P}riddy theorem, Pacific J. Math.. Pacific Journal of Mathematics, 96, 435-455, (1981) · Zbl 0504.55016 [27] Mahowald, Mark, The order of the image of the {$$J$$}-homomorphism. Proc. {A}dvanced {S}tudy {I}nst. on {A}lgebraic {T}opology, {V}ol. {II}, 376-384, (1970) [28] May, J. Peter, Matric {M}assey products, J. Algebra. Journal of Algebra, 12, 533-568, (1969) · Zbl 0192.34302 [29] May, J. Peter, The Cohomology of Restricted Lie Algebras and of Hopf Algebras: Application to the Steenrod Algegra, 171 pp., (1964) [30] Milnor, John, On manifolds homeomorphic to the {$$7$$}-sphere, Ann. of Math. (2). Annals of Mathematics. Second Series, 64, 399-405, (1956) · Zbl 0072.18402 [31] Milnor, John, Differential topology forty-six years later, Notices Amer. Math. Soc.. Notices of the American Mathematical Society, 58, 804-809, (2011) · Zbl 1225.01040 [32] Moise, Edwin E., Affine structures in {$$3$$}-manifolds. {V}. {T}he triangulation theorem and {H}auptvermutung, Ann. of Math. (2). Annals of Mathematics. Second Series, 56, 96-114, (1952) · Zbl 0048.17102 [33] Moss, R. Michael F., Secondary compositions and the {A}dams spectral sequence, Math. Z.. Mathematische Zeitschrift, 115, 283-310, (1970) · Zbl 0188.28501 [34] Mahowald, Mark; Tangora, Martin, Some differentials in the {A}dams spectral sequence, Topology. Topology. An International Journal of Mathematics, 6, 349-369, (1967) · Zbl 0166.19004 [35] Mimura, Mamoru; Toda, Hirosi, The {$$(n+20)$$}-th homotopy groups of {$$n$$}-spheres, J. Math. Kyoto Univ.. Journal of Mathematics of Kyoto University, 3, 37-58, (1963) · Zbl 0129.15403 [36] Nakamura, Osamu, Some differentials in the {$${\rm mod}\ 3$$} {A}dams spectral sequence, Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci., 1-25, (1975) · Zbl 0368.55017 [37] Newman, M. H. A., The engulfing theorem for topological manifolds, Ann. of Math. (2). Annals of Mathematics. Second Series, 84, 555-571, (1966) · Zbl 0166.19801 [38] Perelman, Grisha, The entropy formula for the {R}icci flow and its geometric applications, (2002) · Zbl 1130.53001 [39] Pontryagin, L. S., Homotopy classification of the mappings of an {$$(n+2)$$}-dimensional sphere on an {$$n$$}-dimensional one, Doklady Akad. Nauk SSSR (N.S.), 70, 957-959, (1950) · Zbl 0035.11101 [40] Priddy, Stewart B., Koszul resolutions, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 152, 39-60, (1970) · Zbl 0261.18016 [41] Quillen, Daniel, The {A}dams conjecture, Topology. Topology. An International Journal of Mathematics, 10, 67-80, (1971) · Zbl 0219.55013 [42] Ravenel, Douglas C., Complex Cobordism and Stable Homotopy Groups of Spheres, Pure Appl. Math., 121, xx+413 pp., (1986) · Zbl 0608.55001 [43] Rohlin, V. A., On a mapping of the {$$(n+3)$$}-dimensional sphere into the {$$n$$}-dimensional sphere, Doklady Akad. Nauk SSSR (N.S.), 80, 541-544, (1951) [44] Serre, {\relax J-P}, Homologie singuli\“ere des espaces fibr\'”es. {A}pplications, Ann. of Math. (2). Annals of Mathematics. Second Series, 54, 425-505, (1951) · Zbl 0045.26003 [45] Smale, Stephen, Generalized {P}oincar\'e’s conjecture in dimensions greater than four, Ann. of Math. (2). Annals of Mathematics. Second Series, 74, 391-406, (1961) · Zbl 0099.39202 [46] Sullivan, Dennis, Genetics of homotopy theory and the {A}dams conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 100, 1-79, (1974) · Zbl 0355.57007 [47] Tangora, Martin C., On the cohomology of the {S}teenrod algebra, Math. Z.. Mathematische Zeitschrift, 116, 18-64, (1970) · Zbl 0198.28202 [48] Tangora, Martin, Some extension questions in the {A}dams spectral sequence. Proceedings of the {A}dvanced {S}tudy {I}nstitute on the {A}lgebraic {T}opology, {V}ol.{III}, Various Publ. Ser., 13, 578-587, (1970) [49] Tangora, Martin C., Computing the homology of the lambda algebra, Mem. Amer. Math. Soc.. Memoirs of the American Mathematical Society, 58, v+163 pp., (1985) · Zbl 0584.55019 [50] Tangora, Martin, Some homotopy groups mod {$$3$$}. Conference on Homotopy Theory, Notas Mat. Simpos., 1, 227-245, (1975) · Zbl 0334.55014 [51] Toda, Hirosi, Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Stud., 49, v+193 pp., (1962) · Zbl 0101.40703 [52] Whitehead, George W., The {$$(n+2)^{\rm nd}$$} homotopy group of the {$$n$$}-sphere, Ann. of Math. (2). Annals of Mathematics. Second Series, 52, 245-247, (1950) · Zbl 0037.39703 [53] Wang, Guozhen; Xu, Zhouli, The algebraic {A}tiyah-{H}irzebruch spectral sequence of real projective spectra, (2016) [54] Xu, Zhouli, The strong {K}ervaire invariant problem in dimension 62, Geom. Topol.. Geometry & Topology, 20, 1611-1624, (2016) · Zbl 1352.55007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.