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arXiv 0905.2111math.GT
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球面上局部凸曲线空间的同伦与上同调-- I

The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

Nicolau C. Saldanha

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18 pages, 12 figures
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摘要：

光滑曲线$γ: [0,1] \to S^2$是局部凸的，如果它的测地曲率在每一点都是正的。J. A. Little证明了所有局部正曲线$γ$与$γ(0) = γ(1) = e_1$和$γ'(0) = γ'(1) = e_2$的空间有三个连通分支$L_{-1,c}$，$L_{+1}$，$L_{-1,n}$ .已知空间$L_{-1,c}$是可收缩的，但其他两个连通分量的拓扑结构还没有很好地理解。我们研究这些空间的同伦和上同调。特别是，对于$L_{-1} = L_{-1,c} \sqcup L_{-1,n}$，我们证明了$\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$，$\dim H^{2k}(L_{(-1)^{(k+1)}}, \RR) \ge 2$，$π_2(L_{+1})$包含$Z^2$的拷贝，$π_{2k}(L_{(-1)^{(k+1)}})$包含$Z$的拷贝。

英文摘要：

A smooth curve $γ: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $γ$ with $γ(0) = γ(1) = e_1$ and $γ'(0) = γ'(1) = e_2$ has three connected components $L_{-1,c}$, $L_{+1}$, $L_{-1,n}$. The space $L_{-1,c}$ is known to be contractible but the topology of the other two connected components is not well understood. We study the homotopy and cohomology of these spaces. In particular, for $L_{-1} = L_{-1,c} \sqcup L_{-1,n}$, we show that $\dim H^{2k}(L_{(-1)^{k}}, \RR) \ge 1$, that $\dim H^{2k}(L_{(-1)^{(k+1)}}, \RR) \ge 2$, that $π_2(L_{+1})$ contains a copy of $Z^2$ and that $π_{2k}(L_{(-1)^{(k+1)}})$ contains a copy of $Z$.

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