论文标题
双曲体积，mod 2同源性和k柔性
Hyperbolic volume, mod 2 homology, and k-freeness
论文作者
论文摘要
我们表明，如果$ m $是任何封闭的，可定向的双曲线$ 3 $ -Manifold，带有$ {\ rm vol} \ m \ m \ le3.69 $，我们有$ {\ rm dim} \ h_1（m; {\ bf f} _2 _2 _2 _2）\ le7 $。由于Culler和Shalen，这可能被认为是结果的定性改进，因为常数$ 3.69 $大于相当于$ω^2 $的序数，这是一组井井有条的有限量$ 3 $ -Manifolds。我们还表明，如果$ {\ rm vol} \ m \ le 3.77 $，我们有$ {\ rm dim} \ h_1（m; {\ bf f} _2）\ le10 $。 这些结果是一种新方法的应用程序，用于获得封闭的，可定向的双曲线$ 3 $ manifold的量的下限，以使给定的$ k \ ge4 $的$π_1（m）$是$ k $ -free。 Among other applications we show that if $π_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if $π_1(M)$ is $5$-free we have ${\rm vol}\ M>3.77$.
We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with ${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the constant $3.69$ is greater than the ordinal corresponding to $ω^2$ in the well-ordered set of finite volumes of hyperbolic $3$-manifolds. We also show that if ${\rm vol}\ M\le 3.77$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le10$. These results are applications of a new method for obtaining lower bounds for the volume of a closed, orientable hyperbolic $3$-manifold such that $π_1(M)$ is $k$-free for a given $k\ge4$. Among other applications we show that if $π_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if $π_1(M)$ is $5$-free we have ${\rm vol}\ M>3.77$.
