Chapter 11 — Čech Cohomology Framework

Prerequisites: Chapters 1--8 (Part I), Appendix B.

This chapter provides the cohomological language for the theory, carefully separating what holds exactly in the discrete setting from what requires additional topological hypotheses.

11.1 Why Cech Cohomology?#

A principal $G$-bundle $P\to M$ is defined by an open cover $\{U_i\}$ with transition functions $g_{ij}:U_i\cap U_j\to G$ satisfying the Cech cocycle condition:

This is precisely the closure of relational transit: composing transits around a loop returns the identity. The moment we choose a gauge (local frame), the language of Cech cohomology becomes intrinsic to the theory.

11.2 Cech Cohomology: Definitions#

Cochain groups#

For a cover $\mathcal{U}=\{U_i\}_{i\in I}$ and coefficient group $G$:

Each element assigns a $g_{i_0\cdots i_k}\in G$ to each $(k+1)$-fold intersection.

Coboundary#

Cohomology#

The full Cech cohomology is:

Geometric meaning#

11.3 The Discrete Setting#

What holds exactly#

In the discrete theory (finite graph $\mathcal{G}_t=(V,E_t,W_t)$ with $G$-valued edge labels), the following are well-defined without additional hypotheses:

1. Edge group elements $g_t(i,j)\in G$ serve as discrete 1-cochains (transition functions).

2. Triangle holonomies $\Omega_t(i,j,k)=g_t(i,j)\,g_t(j,k)\,g_t(k,i)$ serve as the discrete curvature (obstruction to the cocycle condition).

3. Simplicial complex: the clique complex (or flag complex) of the graph provides a canonical finite simplicial complex $K$ whose $k$-simplices are the $(k+1)$-cliques of $\mathcal{G}_t$.

4. Cohomology of $K$: for any abelian coefficient group $A$ (e.g., $\mathbb{Z}$, $\mathbb{R}$, $U(1)$), the simplicial cohomology $H^*(K;A)$ is computable by finite linear algebra and is a well-defined invariant of the graph.

5. Relative cohomology: given a fruit $F$ and its door subcomplex $L\subset K$ (the subcomplex generated by simplices containing door nodes), the pair $(K_F,L)$ yields a well-defined relative cohomology $H^*(K_F,L;A)$.

What requires additional hypotheses#

The following require a continuous enrichment hypothesis (i.e., the assumption that the discrete graph approximates a continuous space):

1. Interpreting $\check H^k$ as a topological invariant of an underlying manifold $M$: this requires specifying $M$ and a triangulation/cover correspondence.

2. Uhlenbeck-type bubbling analysis: the singular set $S$ as a subset of a manifold requires the manifold hypothesis.

3. Excision / deformation retract arguments: relating $H^*(K,L)$ to $H^*(M\setminus S)$ requires the topological bridge (see Section 11.5).

4. Persistent cohomology of fractal singular sets: the filtration by energy requires a continuous parameter space.

11.4 Cohomological Interpretation of Doors#

Energy-based doors (fully discrete)#

The door set $\Sigma_\tau(F,t)=\{i\in F:b_{F,t}(i)\ge\tau\}$ is a subset of vertices of the simplicial complex $K_F$. The subcomplex $L$ generated by door nodes and their incident simplices is well-defined.

The relative cohomology $H^*(K_F,L;\mathbb{Z})$ measures "the topology of the fruit with doors removed" in a purely combinatorial sense.

Continuous interpretation (requires manifold hypothesis)#

Under the assumption that $K_F$ triangulates a compact manifold $M_F$:

• $\Sigma$ corresponds to a finite set of points in $M_F$.
• $M_F\setminus\Sigma$ is the "punctured fruit".
• $H^*(M_F\setminus\Sigma;\mathbb{Z})\cong H^*(K_F,L;\mathbb{Z})$ provided $L$ is a good neighbourhood of $\Sigma$ (excision).

11.5 Discrete--Continuous Bridge#

Hypothesis 11.1 (Continuous enrichment). There exists a compact manifold $M_F$ and a triangulation $\tau:K_F\to M_F$ such that: (a) the 1-skeleton of $K_F$ is isomorphic to the induced graph on $F$, (b) the door nodes in $\Sigma$ correspond to points $\{p_1,\ldots,p_L\}\subset M_F$, (c) there exist open balls $B_\epsilon(p_l)$ with $\bigcup B_\epsilon(p_l)$ deformation-retracting to $\{p_1,\ldots,p_L\}$.

Under Hypothesis 11.1:

Proposition 11.2 (Excision bridge). $H^*(K_F,L;\mathbb{Z})\;\cong\;H^*(M_F,\Sigma;\mathbb{Z})\;\cong\;H^*(M_F\setminus\Sigma;\mathbb{Z}).$

Proof sketch. The triangulation gives $H^*(K_F;\mathbb{Z})\cong H^*(M_F;\mathbb{Z})$. Excision for the pair $(M_F,\bigcup B_\epsilon)$ together with the deformation retract gives the isomorphism. $\square$

Key point. All computations are done on the finite pair $(K_F,L)$ using matrix algebra. The continuous interpretation is an optional layer of geometric meaning.

11.6 World Constitution (Summary Form)#

Combining the discrete and optional continuous layers, the world at time $t$ has the following cohomological data:

11.7 Refinement: Non-Abelian Coefficients#

When $G$ is non-abelian, $\check H^1(X;G)$ classifies principal $G$-bundles but is only a pointed set (not a group). The long exact sequence machinery applies in full only for abelian coefficients.

For computations in this theory:

• Use abelian coefficients ($\mathbb{Z}$, $\mathbb{R}$, $U(1)$) for the three-axis exact sequence.
• Use $G$-valued cochains for bundle classification (Axis 1 at $k=1$).
• The scalar curvature $\omega_t(\triangle)=d_G(\Omega_t,e)^2$ provides fully gauge-invariant numerical data regardless of abelianness.