Consider the set \(L\) of \(3\)-manifolds which can be formed by gluing together the boundaries of two solid tori \(S^1 \times B^2\) using a homeomorphism. Consider the function \(d : L \to \mathbb{N}\) given by the total dimension of the mod-\(2\) homology: \(d(M) = \sum_i \dim H_i(M; \mathbb{Z}_2)\). What is the maximal value of \(d\) on \(L\)?
Solution
1. Every manifold in \(L\) is a closed orientable \(3\)-manifold obtained from a genus-one Heegaard splitting.
It is formed by gluing two oriented solid tori along their boundaries by a homeomorphism, reversing boundary orientation as needed to obtain the closed oriented manifold.
2. Its first homology is cyclic (possibly infinite), so\[ \dim_{\mathbb F_2}H_1(M;\mathbb F_2)\le1. \]
The Mayer–Vietoris sequence for two solid tori shows that \(H_1(M;\mathbb Z)\) is generated by one class; equivalently these manifolds are lens spaces, \(S^1\times S^2\), or \(S^3\). Tensoring and applying the mod-\(2\) universal coefficient theorem gives mod-\(2\) first Betti number at most one.
3. Mod-\(2\) Poincaré duality gives\[ b_0=b_3=1,\qquad b_2=b_1. \]
Every closed connected \(3\)-manifold is mod-\(2\) orientable, so Poincaré duality over \(\mathbb F_2\) applies.
4. Hence\[ d(M)=\sum_i b_i=2+2b_1\le4. \]
Substitute the identities from 3 and the bound \(b_1\le1\) from 2.
5. The bound is attained, for example by \(S^1\times S^2\), which admits a genus-one Heegaard splitting and has \(b_1=b_2=1\) over \(\mathbb F_2\).
Its mod-\(2\) homology has dimension one in each degree \(0,1,2,3\).
6. Therefore\[ \boxed{\max_{M\in L}d(M)=4.} \]
Combine 4 and 5.