tmf所隐含的RP ^n的非浸入，再讨论
Nonimmersions of RP^n implied by tmf, revisited
摘要：
在2002年的一篇论文中，作者和Bruner利用新的谱tmf得到了实射影空间的一些新的非浸入。在本文中，我们完成/纠正了该论文中的两个疏忽。首先要注意的是，在那篇论文中，一个一般的非浸入结果被陈述，它产生了RP^n的新的非浸入，其中n小到48，然而在那里，第一个新的结果出现在n=1536时。在这里，我们给出了一个简单的证明这些被忽视的结果。其次，我们填补了2002年论文证明中的一个空白。那里声称轴向映射f必须满足f^*（X）=X_1+X_2。我们最近意识到这一点并不清楚。然而，在这里，我们表明，它是真的乘一个单位在适当的环，所以我们检索所有的nonimmersion结果声称在原来的文件。最后，我们给出了正维数下tmf^{8*}（RP^ \infty \times RP^ \infty）和tmf^*（CP^ \infty \times CP^ \infty）的完全确定。
英文摘要：
In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions.
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