\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: Exercises

Exercises

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These exercises were curated by Fraser Binns.

exercise 0.0.1

What is \(\Sym^2(S^1)\)?

exercise 0.0.2

Let \((\Sigma, \{\alpha_i\}_{i=1}^g, \{\beta_i\}_{i=1}^g)\) be a Heegaard diagram for \(M\). Let \(A\) be the \(g\times g\) matrix whose entries over \(\ZZ/2\ZZ\) are given by \[a_{ij}=\alpha_{i}\cap \beta_j \text{\;\;mod \;\;} 2.\] Given the matrix \(A\), compute \(H^\bullet(M, \ZZ/2\ZZ)\).

exercise 0.0.3

  1. Let \(\Sigma\) be a surface. Show that \(H_1(\Sym^g(\Sigma))\cong H_1(\Sigma)\).
  2. Show that \(H_1(\Sym^g(\Sigma))/(H_1(T_\alpha)\oplus H_1(T_\beta)\cong H_1(\Sigma))/\langle\{\alpha_i,\beta_j\}_{i,j}\rangle\)
  3. What is the homology class of the boundary of a psuedoholomorphic disk?

exercise 0.0.4

A Lens space \(L(p,q)\) can be defined as the \(3\)-manifold with a Heegaard splitting of genus 1, with an \(\alpha\)-curve given by a longitude of the the torus, and the \(\beta\)-curve in the class \((p, q)\). Use the above obstruction to compute \(\widehat{HHF}^\bullet(L(p,q))\).
figure 0.0.5:The Heegaard diagram for the Lens space \(L(2,3)\)

exercise 0.0.6

Compute the Heegaard Floer homology of the manifold with Heegaard splitting as shown, from the diagram. What is the manifold?

exercise 0.0.7

  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 \(\widehat{HHF}^\bullet(Y)\).
  2. Let \(Y\) be a rational homology sphere. Show that \[\chi(\widehat{HHF}^\bullet(Y)):=\operatorname{rank}(\widehat{HHF}^\bullet_0(Y))-\operatorname{rank}(\widehat{HHF}^\bullet_1(Y))=|H_1(Y;\ZZ)|.\]
  3. 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

exercise 0.0.8

  1. Consider the ``decatigorification" of \(\widehat{HHF}^\bullet\) given by the Euler characteristic as described in the previous question. Show that it is invariant under Heegaard moves. How might the proof of invariance of \(\widehat{HHF}^\bullet\) differ?
  2. Suppose \(Y_1,Y_2\) are \(3\)-manifolds. How are \(\widehat{HHF}^\bullet(Y_1\# Y_2),\widehat{HHF}^\bullet(Y_1),\) and \(\widehat{HHF}^\bullet(Y_2)\) related?
  3. Show that \(\widehat{HHF}^\bullet\) is invariant under stabilization and Hamiltonian isotopy in the sense that the resulting groups are isomorphic.