\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
exercise 0.0.1
- Let \(Y\) be a \(3\)-manifold with Heegaard splitting \((\Sigma,\mathbf{\alpha},\mathbf{\beta})\). Pick orientations on \(\Sigma,\alpha,\beta\). Explain how this induces a \(\ZZ/2\) grading on \(\widehat{HHF}^\bullet(Y)\).
- Let \(Y\) be a rational homology sphere. Show that
\[\chi(\widehat{HHF}^\bullet(Y)):=\operatorname{rank}(\widehat{HHF}^\bullet_0(Y))-\operatorname{rank}(\widehat{HHF}^\bullet_1(Y))=|H_1(Y;\ZZ)|.\]
- A rational homology sphere (i.e. a 3 manifold with \(H_*(Y)\cong H_*(S^3)\)) is called an L-space, if \(\operatorname{rank}(\widehat{HHF}^\bullet(Y))\) is minimal in the sense that \(\operatorname{rank}(\widehat{HHF}^\bullet(Y))=|H_1(Y)|\). Explain why Lens spaces are \(L\)-spaces