In all dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove the existence of closed orientable hyperbolic manifolds that do not admit any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\text {spin}^c$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>spin</mtext> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation> structure, and in fact we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel–Whitney class <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$w_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> and are all arithmetic of simplest type. More generally, we show that for each <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n \ge 4k+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exist infinitely many commensurability classes of closed orientable hyperbolic n-manifolds M with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$w_{4k-1}(M) \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mrow> <mn>4</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Closed hyperbolic manifolds without $\text {spin}^c$ structures

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发表日期：

2026年1月19日

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Jacopo G. Chen

摘要

In all dimensions $n \ge 5$ $n \geq 5$ , we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text {spin}^c$ ${spin}^{c}$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel–Whitney class $w_3$ $w_{3}$ and are all arithmetic of simplest type. More generally, we show that for each $k \ge 1$ $k \geq 1$ and $n \ge 4k+1$ $n \geq 4 k + 1$ , there exist infinitely many commensurability classes of closed orientable hyperbolic n-manifolds M with $w_{4k-1}(M) \ne 0$ $w_{4 k - 1} (M) \neq 0$ .

Closed hyperbolic manifolds without \(\text {spin}^c\) structures

摘要