走向$Ω^lS^{n+l}$中球形类的Browder定理
Towards a Browder theorem for spherical classes in $Ω^lS^{n+l}$
摘要：
Browder证明了若$4n+2\neq 2^{t+1}-2$，则$(4n+2)$维流形$M^{4n+2}$的协边类的Kervaire不变量为零，且$M^{2^{t+1}-2}$是Kervaire不变量1当且仅当$h_t^2\in\mathrm{Ext}(\mathbb{Z}/2,\mathbb{Z}/2)$是永久圈.另一方面，根据Madsen，如果$4n+2\neq 2^t-2$，则$M^{4n+2}$与球面协边（因此是Kervaire不变零），而$M^{2^{t+1}-2}$与球面不协边（因此是Kervaire不变一）当且仅当某些元素$p_{2^{t}-1}^2\in H_*QS^0$是球面的。此外，已知$p_{2^t-1}^2$是球面的当且仅当$h_t^2$是Adams谱序列中的永久循环。此外，具有$2n+1\neq 2^t-1$的类$p_{2n+1}^2\in H_*QS^0$很容易从球形中消除。因此，布劳德定理承认在$H_*QS^0$中某些正方形类是球面的陈述和证明（另见Akhmetev和Eccles的工作）。本文考虑用$n>0$和$4\leqslant l\leqslant +\infty$确定球类$H_*Ω^lS^{n+l}$的问题.我们证明：（1）如果$ξ^2\in H_*Ω^lS^{n+l}$是由$\dimξ+1\neq 2^t$、$\dim ξ+1\equiv 2\textrm{ mod }4$和$n>l$给出的，则$ξ^2$不是球形的。我们称之为广义Browder定理。我们还提出了一些部分结果退化的情况下，对应于$\dimξ\neq 2^t-1$，当$l>n$。(2)对于$l\in\{4,5,6,7,8\}$，$H_*Ω^lS^{n+l}$中唯一的球形类来自于包含底细胞，或Hopf不变元素。这验证了埃克尔斯猜想时，限制到有限循环空间与$l<9$。
英文摘要：
According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant of the cobordism class of a $(4n+2)$-dimensional manifold $M^{4n+2}$ vanishes and $M^{2^{t+1}-2}$ is of Kervaire invariant one if and only if $h_t^2\in\mathrm{Ext}(\mathbb{Z}/2,\mathbb{Z}/2)$ is a permanent cycle. On the other hand, according to Madsen if $4n+2\neq 2^t-2$ then $M^{4n+2}$ is cobordant to a sphere (hence of Kervaire invariant zero) and $M^{2^{t+1}-2}$ is not cobordant to a sphere (hence of Kervaire invariant one) if and only if certain element $p_{2^{t}-1}^2\in H_*QS^0$ is spherical. Moreover, it is known that $p_{2^t-1}^2$ is spherical if and only if $h_t^2$ is a permanent cycle in the Adams spectral sequence. Moreover, classes $p_{2n+1}^2\in H_*QS^0$ with $2n+1\neq 2^t-1$ are easily eliminated from being spherical. Hence, Browder's theorem admits a presentation and proof in terms of certain square classes being spherical in $H_*QS^0$ (see also work of Akhmetev and Eccles). In this note, we consider the problem of determining spherical classes $H_*Ω^lS^{n+l}$ with $n>0$ and $4\leqslant l\leqslant +\infty$. We show (1) if $ξ^2\in H_*Ω^lS^{n+l}$ is given with $\dimξ+1\neq 2^t$ and $\dim ξ+1\equiv 2\textrm{ mod }4$ and $n>l$, then $ξ^2$ is not spherical. We refer to this as a generalised Browder theorem. We also present some partial results on the degenerate cases, corresponding to $\dimξ\neq 2^t-1$, when $l>n$. (2) For $l\in\{4,5,6,7,8\}$ the only spherical classes in $H_*Ω^lS^{n+l}$ arise from the inclusion of the bottom cell, or the Hopf invariant one elements. This verifies Eccles conjecture when restricted to finite loop spaces with $l<9$.
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