Using a Toda bracket computation due to Daniel C Isaksen, we investigate the 45-stem more thoroughly. We prove that theta(2)(4) = 0 using a 4-fold Toda bracket. By work of Barratt, Jones and Mahowald, this implies that theta(5) exists and there exists a theta(5) such that 2 theta(5) = 0. Based on theta(2)(4) = 0, we simplify significantly their 9-cell complex construction to a 4-cell complex, which leads to another proof that theta(5) exists.
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机译：由于使用了Daniel C Isaksen的Toda方括号计算，我们更加彻底地研究了45个词干。我们使用4倍Toda括号证明theta（2）（4）= 0。通过Barratt，Jones和Mahowald的工作，这意味着存在theta（5）并且存在一个theta（5）使得2 theta（5）=0。基于theta（2）（4）= 0，我们进行了显着简化他们将9个单元的复合体构造为4个单元的复合体，这导致了theta（5）存在的另一种证明。
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