I am new to algebraic topology and have no experience at all in this field. I started reading Hatchers Book on algebraic topology, however i am still not able to fully understand the following paper by Leo Grady http://leogrady.net/wp-content/uploads/2017/01/grady2010minimal.pdf. More specifically i have trouble understanding page 6. I want to apply the results obtained in the paper linked to a polyhedral complex $K$ that is given as follows:
Imagine a cuboid and randomly choose points in its interior. Consider a Voronoi Diagram inside the cuboid using the standard euclidean norm. This procedure yields a polyhedral complex with the following elements:
- The 0 cells ($C_0(K)$) inside this complex correspond to the vertices of the resulting polyhedrons
- The 1 cells ($C_1(K)$) correspond to the 1 dimensional faces (edges) of the resulting polyhedrons
- The 2 cells ($C_2(K)$) are the facets of the resulting polyhedrons which we consider twice in 2 different orientations
- The 3 cells ($C_3(K)$) are the resulting polyhedrons
Orient the edges arbitrarily but fixed. Consider each 2-dimensional face in 2 orientations. The 3-dimensional cells consist of the facets of the resulting polyhedron in both orientations.
I understood why the matrix $B$ defined as $B_{i,j} := \begin{cases} 1 & \text{if edge } a_i \text{ and facet } f_j \text{ are incident and coherent,}\\ -1 & \text{if edge } a_i \text{ and facet }f_j \text{ are incident and anti-coherent,}\\ 0 & \text{else}\\ \end{cases} $
is the boundary operator $\partial^K_2$ of the following chain complex:
$\cdot\cdot\cdot \xrightarrow{\partial^K_4} C_{3}(K) \xrightarrow{\partial^K_3} C_{2}(K) \xrightarrow{\partial^K_2} C_{1}(K) \xrightarrow{\partial^K_1} C_{0}(K) \longrightarrow 0$
Now, i have the following question:
- How does the incidence matrix C corresponding to $\partial^K_3$ look like? According to the paper i thought of something like this: $C \in \{0,-1,1\}^{2 \cdot \#facets \times \#polyhedrons}$ , where $C_{i,j} := \begin{cases} 1 & \text{if facet } f_i \text{ and polyhedron } c_j \text{ are incident and coherent,}\\ -1 & \text{if facet } f_i \text{ and polyhedron } c_j \text{ are incident and anti-coherent,}\\ 0 & \text{else}\\ \end{cases} $ where the 3 dimensional cells are chosen such that two adjacent cells have opposite orientations w.r.t. any common facet. How can i prove that such a complex has trivial homology? I need Ker($B$) $=$ Im($C$) in order for the subsequent steps in the paper to hold true.
I would be happy if any of you could recommend literature on cell complexes in the context of subdivisons of compact spaces with the focus being on complexes with trivial homology. I want to better understand boundary operators and how to compute them. What puzzled me the most in the above mentioned papers is that the same facet was considered twice in 2 different orientations. I had trouble working out how this special case fits within the framework of algebraic topoogy.