- Cellular Paving

In the previous examples, cells fully paved the two-dimensional plane without any gap. Such a paving is often called a mesh in finite element analysis. Naturally, a *p-cell* is defined as a *p*-dimensional directed face, which is extended from a 0-cell (point), 1-cell (edge), and 2-cell (face). Cellular paving with the cells can be rigorously formulated in higher-dimensional spaces. The boundary of each element is represented by a combination of lower-dimensional cells. Strictly speaking, a cellular paving of some region *R* in a manifold is a finite set of open directed *p*-cells such that (i) two distinct cells do not intersect, (ii) the union of all cells is *R*, and (iii) if the closures of two cells *c* and *c* meet, their intersection is the closure of a unique cell *c* [62]. However, we only need to grasp the concept of cellular paving with an intuitive sense.

For a cellular paving K in an *m*-dimensional region, we can define *Ci*(K) (*i* = 0, 1, ··· , *m*) generated from *i*-cells with a boundary operator *∂* : *Cp*(K) → *Cp*−<sup>1</sup>(K) satisfying *∂* ◦ *∂* = 0. Following a similar treatment for graphs, *Zp*(K) = ker *∂* : *Cp*(K) → *Cp*−<sup>1</sup>(K) and *Bp*(K) = im *∂* : *Cp*+<sup>1</sup>(K) → *Cp*(K) are defined. Next, we consider a dual space *Cp*(K)=( *Cp*(K))<sup>∗</sup>. As dual counterparts of *Zp*(K) and *Bp*(K), we can define *Zp*(K) = ker d : *Cp*(K) → *Cp*+<sup>1</sup>(K) and *Bp*(K) = im d : *Cp*−<sup>1</sup>(K) → *Cp*(K) . To extract topological characteristics, homology and cohomology groups are defined as *Hp*(K) = *Zp*(K)/*Bp*(K) and *Hp*(K) = *Zp*(K)/*Bp*(K), respectively. It is known that dim *Hp*(K) and dim *Hp*(K) do not depend on specific cellular paving, but only on the original *m*-dimensional region.

#### *2.3. Inner and Outer Orientations*

#### - Intuitive Explanation of Inner and Outer Orientations

Thus far, cells were assumed to be (internally) oriented. In this subsection, we clarify the definition of orientation and provide a deeper explanation that is crucial for understanding the circuit duality. First, one can intuitively grasp two types of orientation defined for a surface in a three-dimensional space. In Figure 7a, the surface is internally oriented, where "internally" means that the orientation is defined through the internal coordinates of the surface, regardless of whether or not the surface is embedded in the three-dimensional space. On the other hand, we could consider an outer orientation of the surface as the normal-vector field as shown in Figure 7b. Outer orientation refers to the transverse direction of the surface and involves ambient space. To discuss how much fluid traverses the surface, we do not use an inner-oriented surface, but an outer-oriented one. These two types of orientation are strictly defined below and are important to express various physical quantities.

**Figure 7.** Two types of orientation: (**a**) inner and (**b**) outer orientation of a surface in a *three-dimensional* space.

#### - Inner Orientation of Vector Space and Cell

As a starting point, we define an inner orientation of a vector space *U*. Consider two ordered bases (*<sup>e</sup>*1, *e*2, ··· ,*em*) and (*e*1, *e*2, ··· ,*em*) in *U*. A basis transformation is represented by *ei* = ∑*mj*=<sup>1</sup> *ejPji*. When det *P* > 0, the direction associated with (*<sup>e</sup>*1, *e*2, ··· ,*em*) is considered to be the same as that with (*e*1, *e*2, ··· ,*em*); otherwise, their directions are different. Then, bases are classified as two different equivalence classes. We can choose one specific class to be positive orientation. If a vector space *U* has such a positive orientation, *U* is called inner-oriented. Generally, a tangent space T*PM* is a *p*-dimensional vector space composed of tangent vectors at a point *P* on a *p*-dimensional surface (manifold) (Figure 8a). If tangent spaces on a *p*-cell are continuously oriented, it is called inner-oriented. See a schematic picture shown in Figure 8b.

**Figure 8.** (**a**) tangent space T*PM* at a point *P* on a surface *M*; (**b**) inner-oriented 2-cell with continuously oriented tangent spaces.

#### - Outer Orientation of Vector Space and Cell

Next, we introduce an outer orientation. Consider a linear subspace *W* in a vector space *U*. When we choose an orientation of the quotient vector space *U*/*W*, we say that *W* is *outer-oriented*. The outer orientation is naturally extended to a *p*-cell *S* in an *m*-dimensional manifold *M*. Here, *M* is a total space including *S* and called ambient space. Because of T*PS* ⊂ T*PM*, we can consider T*PM*/T*PS* as a vector space whose (nonzero) elements are considered to be transverse to T*PS*. If a *p*-cell has continuous orientation of T*PM*/T*PS* for all *P* ∈ *S*, the cell is called outer-oriented. An example of an outer-oriented 2-cell is shown in Figure 9.

**Figure 9.** Outer-oriented 2-cell *S* in a three-dimensional space is continuously outer-oriented in all tangent spaces.

For a planar paving, we summarize all types of cells in Figure 10. An outer-oriented 0-cell is best suited to represent a rotation around a point in the two-dimensional plane. Fluid flow transverse to an edge is also represented by an outer-oriented 1-cell.


**Figure 10.** Inner- and outer-oriented cells in a *two-dimensional plane*.


Outer orientation can be represented by two inner orientations depending on the orientation of ambient space. Let us see this representation in an example in a two-dimensional plane. Consider an outer-oriented edge in a planar graph. With a given orientation of the plane (two-dimensional Euclid space E2), we can convert the outer orientation into an inner one as shown in Figure 11. The inner orientation is induced by rotating the outer orientation by 90◦, where the rotational direction is determined by the ambient-space orientation. Importantly, the inner direction is reversed when we choose the opposite ambient-space orientation. Thus, an outer-oriented cell *s*ˇ can be represented by two different inner-oriented cells as *s*ˇ = {*s*<sup>ˇ</sup>*o*|*<sup>o</sup>* <sup>=</sup>, } with *<sup>s</sup>*<sup>ˇ</sup> = <sup>−</sup>*s*<sup>ˇ</sup>. The above intuitive discussion can be generalized in other dimensional spaces as shown in Appendix A.

**Figure 11.** Inner-oriented components of an outer-oriented edge in a two-dimensional plane. The plane orientations are given by the subscripts (, ).


The previous discussions on orientations are based on representations of an outer-oriented cell by inner-oriented cells. We would like to remark on a dual perspective: an inner-oriented cell can be represented by outer-oriented cells. This perspective is indicated in Figure 12. To obtain an outer orientation, the inner orientation of the edge is rotated by 90◦ in the *reverse* direction of a given orientation of the plane.

**Figure 12.** Outer-oriented components of an inner-oriented edge in a two-dimensional plane.


Next, consider a cellular paving Kˇ with outer-oriented cells. We can define a vector space *Cp*(K<sup>ˇ</sup> ) generated from *p*-cells in Kˇ . Using the representation of an outer-oriented cell by inner-oriented cells, we can define *∂* : *Cp*(K<sup>ˇ</sup> ) → *Cp*−<sup>1</sup>(K<sup>ˇ</sup> ) so that *∂* individually acts for each inner-oriented component. For example, an inner-oriented component of the boundary of an outer-oriented 1-cell in a plane is depicted in Figure 13. The boundary operations for outer-oriented cells in a two-dimensional plane are summarized in Figure 14.

**Figure 13.** Boundary operation for an outer-oriented 1-cell in a two-dimensional plane is defined through the inner-oriented component.

**Figure 14.** Boundary operation for an outer-oriented (**a**) edge and (**b**) face in a two-dimensional plane.

As a dual counterpart of *Cp*(K<sup>ˇ</sup> ), the dual space *Cp*(K<sup>ˇ</sup> )=( *Cp*(K<sup>ˇ</sup> ))∗ is defined by applying the general theory for linear spaces. An outer-oriented cochain in *Cp*(K<sup>ˇ</sup> ) is also represented by two inner-oriented cochains and the coboundary operator d for outer-oriented cochains is naturally defined.

#### *2.4. Essence of Poincaré Duality*
