45 Def 27: Horizontal/Vertical Homology Splitting
Let \(H\) be a finite group acting on a graph \(X\) with cell labeling \(\Lambda \) that is invariant under the action. The Tanner code \(H\)-equivariant chain complex is defined as
the \(H\)-equivariant chain complex associated to the Tanner code on the graph with group action \(X\) and labeling \(\Lambda \).
Let \(\ell \geq 3\) be an odd natural number with \(H\) acting on \(\operatorname {Fin}(\ell )\) via a cycle-compatible action \(h_{\mathrm{compat}}\). The cycle graph \(H\)-equivariant chain complex is defined as
the \(H\)-equivariant chain complex associated to the cycle graph \(C_\ell \).
The balanced product complex \(C(X, \Lambda ) \otimes _H C(C_\ell )\) is defined as
the balanced product of the Tanner code complex and the cycle graph complex over \(H\).
The degree-1 homology of the balanced product complex is defined as
the first homology group of \(C(X, \Lambda ) \otimes _H C(C_\ell )\) over \(\mathbb {F}_2\).
The horizontal homology is the balanced Künneth summand at indices \(p = 1\), \(q = 0\):
The vertical homology is the balanced Künneth summand at indices \(p = 0\), \(q = 1\):
Assuming \(|H|\) is odd, the Künneth isomorphism at degree \(1\) is the \(\mathbb {F}_2\)-linear equivalence
defined as \(\phi := \operatorname {kunnethBalancedProduct}\! \bigl(\operatorname {tannerHEq}(X, \Lambda , h_\Lambda ),\; \operatorname {cycleHEq}(\ell , h_{\mathrm{compat}}),\; h_{\mathrm{odd}},\; 1\bigr)\).
For each integer \(p \in \mathbb {Z}\), the canonical projection onto the \(p\)-th summand is the \(\mathbb {F}_2\)-linear map
defined as \(\pi _p := \operatorname {DirectSum.component}_{\mathbb {F}_2}(-)(p)\).
For each integer \(p \in \mathbb {Z}\), the canonical inclusion of the \(p\)-th summand into the direct sum is the \(\mathbb {F}_2\)-linear map
defined as \(\iota _p := \operatorname {DirectSum.lof}_{\mathbb {F}_2}(-)(p)\).
The horizontal projection \(p^h : H_1 \to H_1^h\) is the \(\mathbb {F}_2\)-linear map defined as the composition of the Künneth isomorphism with the canonical projection onto the \(p = 1\) summand:
where \(\phi : H_1 \xrightarrow {\sim } \bigoplus _p H_p \otimes _H H_{1-p}\) is the Künneth isomorphism and \(\pi _1\) is the projection onto the \(p = 1\) component.
The vertical projection \(p^v : H_1 \to H_1^v\) is the \(\mathbb {F}_2\)-linear map defined as the composition of the Künneth isomorphism with the canonical projection onto the \(p = 0\) summand:
where \(\phi : H_1 \xrightarrow {\sim } \bigoplus _p H_p \otimes _H H_{1-p}\) is the Künneth isomorphism and \(\pi _0\) is the projection onto the \(p = 0\) component.
A homology class \(x \in H_1\) is horizontal if its vertical projection vanishes:
Equivalently, under the Künneth isomorphism, \(x\) lies entirely in the horizontal summand \(H_1^h \oplus \{ 0\} \).
A homology class \(x \in H_1\) is vertical if its horizontal projection vanishes:
Equivalently, under the Künneth isomorphism, \(x\) lies entirely in the vertical summand \(\{ 0\} \oplus H_1^v\).
The horizontal cohomology is the balanced Künneth summand at indices \(p = 1\), \(q = 0\):
By the homology-cohomology equivalence over \(\mathbb {F}_2\), this coincides with \(H_1^h\).
The vertical cohomology is the balanced Künneth summand at indices \(p = 0\), \(q = 1\):
By the homology-cohomology equivalence over \(\mathbb {F}_2\), this coincides with \(H_1^v\).
The cohomology horizontal projection \(p_h : H^1 \to H^1_h\) is defined as
identifying \(\operatorname {coprojH}\) with \(\operatorname {projH}\) via the homology-cohomology equivalence over \(\mathbb {F}_2\).
The cohomology vertical projection \(p_v : H^1 \to H^1_v\) is defined as
identifying \(\operatorname {coprojV}\) with \(\operatorname {projV}\) via the homology-cohomology equivalence over \(\mathbb {F}_2\).
For every pair of elements \(y_h \in H_1^h\) and \(y_v \in H_1^v\), there exists \(x \in H_1\) such that
Let \(\phi : H_1 \xrightarrow {\sim } \bigoplus _p H_p \otimes _H H_{1-p}\) be the Künneth isomorphism. We use the inverse \(\phi ^{-1}\).
Construct the element
where \(\iota _1\) and \(\iota _0\) are the canonical inclusions of the \(p = 1\) and \(p = 0\) summands respectively. Set \(x := \phi ^{-1}(z)\).
We verify both projections. For the horizontal projection, simplifying using the definitions of \(\operatorname {projH}\), \(\pi _1\), \(\iota _1\), \(\iota _0\), and the fact that \(\phi \circ \phi ^{-1} = \operatorname {id}\):
By the defining properties of direct sum components, \(\pi _1(\iota _1(y_h)) = y_h\) and \(\pi _1(\iota _0(y_v)) = 0\) since \(0 \neq 1\) (verified by integer arithmetic). Hence \(p^h(x) = y_h\).
For the vertical projection, by the same reasoning with \(\pi _0\):
since \(1 \neq 0\) (verified by integer arithmetic) implies \(\pi _0(\iota _1(y_h)) = 0\), and \(\pi _0(\iota _0(y_v)) = y_v\). Thus \(p^v(x) = y_v\).
The type \(H_1^h\) is nonempty; it contains the zero element.
The zero element \(0 \in H_1^h\) witnesses nonemptiness.
The type \(H_1^v\) is nonempty; it contains the zero element.
The zero element \(0 \in H_1^v\) witnesses nonemptiness.
The type \(H_1\) is nonempty; it contains the zero element.
The zero element \(0 \in H_1\) witnesses nonemptiness.