This paper contains some results about the topology of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {M}_{0,n+1}/\Sigma _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {M}_{0,n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the moduli space of genus zero Riemann surfaces with marked points. We show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {M}_{0,n+1}/\Sigma _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is not a topological manifold for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and it is simply connected for any <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. We also present some homology computations: for example we show that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {M}_{0,p+1}/\Sigma _p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> has no p torsion, where p is a prime. Lastly we compute <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$H_*(\mathcal {M}_{0,n+1}/\Sigma _n;\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>H</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> <mo>;</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for small values of n, proving that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {M}_{0,n+1}/\Sigma _n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is contractible for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n\le 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> while <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {M}_{0,7}/\Sigma _6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">M</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi mathvariant="normal">Σ</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is not.

On the topology of $\mathcal {M}_{0,n+1}/\Sigma _n$

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2025年11月2日

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Tommaso Rossi

摘要

This paper contains some results about the topology of $\mathcal {M}_{0,n+1}/\Sigma _n$ $M_{0, n + 1} / Σ_{n}$ , where $\mathcal {M}_{0,n+1}$ $M_{0, n + 1}$ is the moduli space of genus zero Riemann surfaces with marked points. We show that $\mathcal {M}_{0,n+1}/\Sigma _n$ $M_{0, n + 1} / Σ_{n}$ is not a topological manifold for $n\ge 4$ $n \geq 4$ , and it is simply connected for any $n\in \mathbb {N}$ $n \in N$ . We also present some homology computations: for example we show that $\mathcal {M}_{0,p+1}/\Sigma _p$ $M_{0, p + 1} / Σ_{p}$ has no p torsion, where p is a prime. Lastly we compute $H_*(\mathcal {M}_{0,n+1}/\Sigma _n;\mathbb {Z})$ $H_{*} (M_{0, n + 1} / Σ_{n}; Z)$ for small values of n, proving that $\mathcal {M}_{0,n+1}/\Sigma _n$ $M_{0, n + 1} / Σ_{n}$ is contractible for $n\le 5$ $n \leq 5$ while $\mathcal {M}_{0,7}/\Sigma _6$ $M_{0, 7} / Σ_{6}$ is not.

On the topology of \(\mathcal {M}_{0,n+1}/\Sigma _n\)

摘要