How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:43:36Z http://mathoverflow.net/feeds/question/101540 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101540/how-do-you-know-when-something-must-die-in-the-adams-spectral-sequence-for-pi How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$ Joseph Victor 2012-07-06T23:21:06Z 2012-07-07T04:42:34Z <p>Hey everybody,</p> <p>I think this question might be just a simple oversight on my part, but this has been bugging me a few days. </p> <p>I am reading Hatcher's <a href="http://www.math.cornell.edu/~hatcher/SSAT/SSch2.pdf" rel="nofollow">Spectral Sequences book</a>, and trying to understand his example where he computes $\pi_*^s$ for $p=2$ (page 21-23), and I'm a bit confused about a certain step. He claims that the element corresponding to $h_3^2$ must have order 2 in $\pi_{14}^s$, because of "the commutativity property of the composition product, since $h_3$ has odd degree". Now, I see why $h_3^2$ can have order at most 4, because $h_3^2h_0^2=0 \in E_2$, but why must it have order 2 exactly? What does the odd degree have to do with it? If I am not mistaken, the Yoneda product on $Ext_A(Z/2,Z/2)$ induces the composition product on $\pi_*^s$, which, mod 2, is commutative, but the Yoneda product has $h_3h_0=h_0h_3$ in the $E_2$ page, so I can't from that derive the induced composition product is 0. Do I need to use a fact about $\pi_s^*$ that doesn't come from this spectral sequence? </p> <p>Thanks for the help everybody! -Joseph Victor</p> http://mathoverflow.net/questions/101540/how-do-you-know-when-something-must-die-in-the-adams-spectral-sequence-for-pi/101559#101559 Answer by Tyler Lawson for How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$ Tyler Lawson 2012-07-07T04:42:34Z 2012-07-07T04:42:34Z <p>The point is that the element <code>$h_3$</code> in odd degree lifts to an element (typically called $\sigma$) in <code>$\pi_7^s$</code> whose square must be 2-torsion. This is true for all elements in odd degree because the stable homotopy groups of spheres are graded-commutative. It does not necessarily have to be exactly 2-torsion a priori - it could be 1-torsion (i.e. trivial) - but it must be annihilated by 2.</p>