40 Def 23: Balanced Product Chain Complex
Let \(H\) be a group. An \(H\)-equivariant chain complex over \(\mathbb {F}_2\) consists of:
an underlying chain complex \(C_\bullet \) over \(\mathbb {F}_2\),
for each degree \(i \in \mathbb {Z}\), a representation \(\rho _i : H \to \operatorname {GL}_{\mathbb {F}_2}(C_i)\) (i.e., a left \(\mathbb {F}_2[H]\)-module structure on \(C_i\)),
an equivariance condition: for all \(i, j \in \mathbb {Z}\) and \(h \in H\),
\[ \partial ^{i \to j} \circ \rho _i(h) = \rho _j(h) \circ \partial ^{i \to j}, \]i.e., the differential \(\partial : C_i \to C_j\) intertwines the \(H\)-actions.
Let \(C\) and \(D\) be \(H\)-equivariant chain complexes. For integers \(p, q \in \mathbb {Z}\), the balanced product object at \((p,q)\) is
the balanced product (coinvariants of \(C_p \otimes _{\mathbb {F}_2} D_q\) under the diagonal \(H\)-action) of the representations \(\rho ^C_p\) and \(\rho ^D_q\).
Let \(C\) and \(D\) be \(H\)-equivariant chain complexes. The horizontal differential
is induced by \(\partial ^C_{p \to p'} \otimes \operatorname {id}_{D_q}\) on the balanced product. This is well-defined on coinvariants by the \(H\)-equivariance of \(\partial ^C\).
Let \(C\) and \(D\) be \(H\)-equivariant chain complexes. The vertical differential
is induced by \(\operatorname {id}_{C_p} \otimes \partial ^D_{q \to q'}\) on the balanced product. This is well-defined on coinvariants by the \(H\)-equivariance of \(\partial ^D\).
Let \(C\) and \(D\) be \(H\)-equivariant chain complexes. The balanced product double complex \(C \boxtimes _H D\) is the double complex over \(\mathbb {F}_2\) with:
objects \((C \boxtimes _H D)_{p,q} = C_p \otimes _H D_q\),
horizontal differential \(\partial ^h = \partial ^C \otimes _H \operatorname {id}_D\) (decreasing \(p\)),
vertical differential \(\partial ^v = \operatorname {id}_C \otimes _H \partial ^D\) (decreasing \(q\)).
This forms a valid double complex: \((\partial ^h)^2 = 0\), \((\partial ^v)^2 = 0\), and \(\partial ^h \circ \partial ^v = \partial ^v \circ \partial ^h\).
Let \(C\) and \(D\) be \(H\)-equivariant chain complexes. The balanced product complex
is the total complex of the balanced product double complex \(C \boxtimes _H D\).
For all \(p, q \in \mathbb {Z}\), the object of the balanced product double complex at \((p,q)\) satisfies
This holds by reflexivity, as the definition of \(\operatorname {balancedProductDoubleComplex}\) sets the object at \((p,q)\) to be \(\operatorname {bpObj}(p,q) = C_p \otimes _H D_q\) definitionally.
For all \(p, q \in \mathbb {Z}\), the vertical differential of the balanced product double complex satisfies
i.e., it is induced by \(\operatorname {id}_{C_p} \otimes \partial ^D_{q \to q-1}\).
This holds by reflexivity, as \(\operatorname {dv}\) at \((p,q)\) in the balanced product double complex is defined as the linear map underlying \(\operatorname {bpDv}(p, q, q-1)\).
For all \(p, q \in \mathbb {Z}\), the horizontal differential of the balanced product double complex satisfies
i.e., it is induced by \(\partial ^C_{p \to p-1} \otimes \operatorname {id}_{D_q}\).
This holds by reflexivity, as \(\operatorname {dh}\) at \((p,q)\) in the balanced product double complex is defined as the linear map underlying \(\operatorname {bpDh}(p, p-1, q)\).
For every \(n \in \mathbb {Z}\), the degree-\(n\) component of the balanced product complex \(C \otimes _H D\) is nonempty (as it contains the zero element).
We exhibit the zero element \(0 \in (C \otimes _H D)_n\), which witnesses that the type is nonempty.