论文标题
具有奇数Euler特征的高维歧管的可定义性
Orientability of high-dimensional manifolds with odd Euler characteristic
论文作者
论文摘要
如果$ i^{th} $ stiefel-whitney类消失了所有$ i <2^k $（$ k \ geq 0 $），我们将其称为歧管$ k $ - 方向。在\ cite {hoekzema2017}中，显示出$ k $方向的歧管甚至具有欧拉的特征（实际上消失了wu class），除非它们的尺寸为$ 2^{k+1} m $，对于某些$ M \ geQ 1 $。该定理对$ k = 0,1,2,3 $严格，但是是否存在具有奇数欧拉特征的4个方向流形，这是一个悬而未决的问题。本文讨论了找到这种歧管$ \ mathfrak {x}^{32m} $的候选人的问题。作为调查的一部分，我们研究了三个被称为罗森菲尔德平面的三个特殊对称空间的示例，它们具有奇怪的欧拉特征，具有32、64和128的维度。我们对steenrod代数对这些谱系的Mod 2同学对使用计算机计算的Mod 2同学的作用进行了计算。第一个Rosenfeld平面，$（\ Mathbb {o} \ otimes \ Mathbb {C}）\ Mathbb {p}^{2} $，是2-定向但不是3-方向而不是$ \ Mathfrak {X}}^{32} $的示例。我们表明，第二架Rosenfeld Plane $（\ MATHBB {O} \ otimes \ Mathbb {h}）\ Mathbb {p}^{2} $是3个方向的，我们提供的条件可能是4个方向而定，如果是Steenrod Algebra的动作，并且它仍然是一个潜在的候选，并且它是一个潜在的候选，并且它可能是一个方向的，并且它可能是一种条件。 $ \ mathfrak {x}^{64} $。作者知道$ \ mathfrak {x}^{32m} $的其他明确候选歧管，或者尤其是$ \ mathfrak {x}^{32} $的候选者。
We call a manifold $k$-orientable if the $i^{th}$ Stiefel-Whitney class vanishes for all $i< 2^k$ ($k\geq 0$), generalising the notions of orientable (1-orientable) and spin (2-orientable). In \cite{Hoekzema2017} it was shown that $k$-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is $2^{k+1}m$ for some $m\geq 1$. This theorem is strict for $k=0,1,2,3$, but whether there exist 4-orientable manifolds with an odd Euler characteristic is an open question. This paper discusses the question of finding candidates for such a manifold $\mathfrak{X}^{32m}$. As part of our investigation we study the example of the three exceptional symmetric spaces known as Rosenfeld planes, which have odd Euler characteristic and are of dimension 32, 64 and 128. We perform computations of the action of the Steenrod algebra on the mod 2 cohomology of the first two of these manifolds with the use computer calculations. The first Rosenfeld plane, $(\mathbb{O} \otimes \mathbb{C})\mathbb{P}^{2}$, is 2-orientable but not 3-orientable and thus not an example of $\mathfrak{X}^{32}$. We show that the second Rosenfeld plane $(\mathbb{O} \otimes \mathbb{H})\mathbb{P}^{2}$ is 3-orientable and we present a condition under which is may be 4-orientable if the action of the Steenrod algebra is established further, and therefore it remains a potential candidate for $\mathfrak{X}^{64}$. No other clear candidate manifolds for $\mathfrak{X}^{32m}$ or in particular candidates for $\mathfrak{X}^{32}$ are known to the author.
