Topological Persistence for CircleValued Maps
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Abstract
We study circlevalued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of realvalued maps, circlevalued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circlevalued map on an input simplicial complex.
Keywords
Topological persistence Persistence for circlevalued maps Bar codes Jordan cells1 Introduction
Data analysis provides many scenarios where one ends up with a nice space, most often a simplicial complex, a smooth manifold, or a stratified space equipped with a realvalued or a circlevalued map. The persistence theory, introduced in [14], provides a useful tool for analyzing realvalued maps with the help of homology. A similar theory for circlevalued maps has not yet been developed in the literature. The work in [9] introduces the concept of circlevalued maps in the context of persistence by deriving a circlevalued map for given data by using the existing persistence theory. In contrast, we develop a persistence theory for circlevalued maps.
One place where circlevalued maps appear naturally is the area of dynamics of vector fields. Many dynamics are described by vector fields which admit a minimizing action (in mathematical terms a Lyapunov closed one form). Such actions can be interpreted as 1cocycles which are intimately connected to circlevalued maps as shown in [3]. Consequently, a notion of persistence for circlevalued maps also provides a notion of persistence for 1cocycles which appear in some data analysis problems [20, 21]. In summary, persistence theory for circlevalued maps promises to play the role for some vector fields as does the standard persistence theory for the scalar fields [6, 7, 14, 22].
One of the main concepts of the persistence theory is the notion of bar codes [22]—invariants that characterize a realvalued map at the homology level. The angle (circle) valued maps, when characterized at homology level, require a new invariant called Jordan cells in addition to the refinement of the bar codes into four types.
The standard persistence [14, 22] which we refer as sublevel persistence deals with the change in the homology of the sublevel sets which cannot make sense for a circlevalued map. However, the change in the homology of the level sets can be considered for both real and circlevalued maps. The notion of persistence, when considered for the level sets of a realvalued map [10] is referred here as level persistence. It refines the sublevel persistence. The zigzag persistence introduced in [4] provides complete invariants (bar codes) for level persistence of (tame) realvalued maps. They are defined by using representation theory for linear quivers.
The change in homology of the level sets of a (tame) circlevalued map is more complicated because of the return of the level to itself when one goes along the circle. It turns out that representation theory of cyclic quivers provides the complete invariants for persistence in the homology of the level sets of the circlevalued maps. This notion of persistence is called here the persistence for circlevalued maps and its invariants, bar codes and Jordan cells are shown to be effectively computable.
Our results include a derivation of the homology for the source space and its relevant subspaces in terms of the invariants (Theorems 3.1 and 3.2). The result also applies to realvalued maps as they are special cases of the circlevalued maps. This leads to a result (Corollary 3.4) which to our knowledge has not yet appeared in the literature.^{1} A number of other topological results which cannot be derived from any of the previously defined persistence theories are described in [2] providing additional motivation for this work.
After developing the results on invariants, we propose a new algorithm to compute the bar codes and Jordan cells. For a simplicial complex, the entire computation can be done by manipulating the original matrix that encodes the input complex and the map. The algorithm first builds a block matrix from the original incidence matrix which encodes linear maps induced in homology among regular and critical level sets, more precisely the quiver representations \(\rho _r\) described in Sect. 4. Next, it iteratively reduces this new matrix eliminating and hence computing the bar codes. The resulting matrix which is invertible can be further processed to Jordan canonical form [12] providing Jordan cells. The algorithm for zigzag persistence [4] when applied to what we refer in Sect. 3 as the infinite cyclic covering map \(\tilde{f}\) can compute bar codes but not Jordan cells. In contrast, our method can compute the bar codes and Jordan cells simultaneously by manipulating matrices and can also be used as an alternative to compute the bar codes in zigzag persistence.

For \(r\)th homology group of a topological space X under an a priori fixed field \(\kappa \), we write \(H_r(X)\) instead of \(H_r(X;\kappa )\).

For a map \(f:X\rightarrow Y\) and \(K\subseteq Y\) we write \(X_K:= f^{1}(K)\).

We use \(\mathbb Z _{\ge 0}\) and \(\mathbb Z _{>0}\) for nonnegative and positive integers respectively.

In our exposition, we need to use open, semiopen, and closed intervals denoted as \((a,b)\), \((a,b]\) or \([a,b)\), and \([a,b]\) respectively. To denote an interval, in general, we use the notation \(\{a,b\}\) where “{” stands for either “[” or “(”.
 For a linear map \(\alpha : V\rightarrow W\) between two vector spaces we write:$$\begin{aligned}&\ker \alpha :=\big \{v\in V \mid \alpha (v)=0\big \},\\&\mathrm img \ \alpha := \big \{w\in \alpha (V)\subseteq W\big \},\\&\mathrm coker\ \alpha := W/\alpha (V). \end{aligned}$$

A matrix A is said to be in column echelon form if all zero columns, if any, are on the right to nonzero ones and the leading entry (the first nonzero number from below) of a nonzero column is always strictly below of the leading entry of the next column. Similarly, A is said to be in row echelon form if all zero rows, if any, are below nonzero ones and the leading entry (the first nonzero number from the right) of a nonzero row is always strictly to the right of the leading entry of the row below it. If A is an \(m\times n\) matrix (m rows and n columns), there exist an invertible \(n\times n\) matrix \(R(A)\) and an invertible \(m\times m\) matrix \(L(A)\) so that \(A\cdot R(A)\) is in column echelon form and \(L(A)\cdot A\) is in row echelon form. Algorithms for deriving the column and row echelon form can be found in standard books on linear algebra.
2 Definitions and Background
We begin with the technical definition of tameness of a map.
For a continuous map \(f: X\rightarrow Y\) between two topological spaces X and Y, let \(X_U= f^{1}(U)\) for \(U\subseteq Y\). When \(U=y\) is a single point, the set \(X_y\) is called a fiber over y and is also commonly known as the level set of y. We call the continuous map \(f:X\rightarrow Y\) good if every \(y\in Y\) has a contractible neighborhood U so that the inclusion \(X_y\rightarrow X_U\) is a homotopy equivalence. The continuous map \(f:X\rightarrow Y\) is a fibration if each \(y\in Y\) has a neighborhood U so that the maps \(f: X_U \rightarrow U\) and \(pr: X_y\times U\rightarrow U\) are fiber wise homotopy equivalent. This means that there exist continuous maps \(l:X_U\rightarrow X_y\times U\) with \(pr_U \cdot l_U = f_U\) which, when restricted to the fiber for any \(z\in U\), are homotopy equivalences. In particular, f is good.
Definition 2.1
A proper continuous map \(f:X\rightarrow Y\) is tame if it is good, and for some discrete closed subset \(S\subset Y\), the restriction \(f: X\setminus f^{1}(S) \rightarrow Y \setminus S\) is a fibration. The points in \(S\subset Y\) which prevent f to be a fibration are called critical values.
If \(Y= \mathbb R \) and X is compact or \(Y= \mathbb{S }^1,\) ^{2} then the set of critical values is finite, say \(s_1 < s_2 < \cdots s_k.\) The fibers above them, \(X_{s_i},\) are referred to as singular fibers. All other fibers are called regular. In the case of \(\mathbb{S }^1\), \(s_i\) can be taken as angles and we can assume that \(0< s_i \le 2\pi .\) Clearly, for the open interval \((s_{i1},s_i)\) the map \(f: f^{1}(s_{i1}, s_{i}) \rightarrow (s_{i1}, s_{i})\) is a fibration which implies that all fibers over angles in \((s_{i1}, s_i)\) are homotopy equivalent with a fixed regular fiber, say \(X_{t_i}\), with \(t_i\in (s_{i1}, s_i)\).
In particular, there exist maps \(a_i: X_{t_i}\rightarrow X_{s_i}\) and \(b_i: X_{t_{i+1}}\rightarrow X_{s_{i}}\), unique up to homotopy defined as follows: If \(t_i\) and \(t_{i+1}\) are contained in \(U_i \subset Y\) where the inclusion \(X_{s_i} \subset X_{U_i}\) is a homotopy equivalence with a homotopy inverse \(r_i: X_{U_i} \rightarrow X_{s_i},\) then \(a_i\) and \(b_i\) are the restrictions of \(r_i\) to \(X_{t_i}\) and \(X_{t_{i+1}}\) respectively. If not, in view of the tameness of f, one can find \(t^{\prime }_i\) and \( t^{\prime }_{i+1}\) in \(U_i\) so that \(X_{t_i}\) and \(X_{t_{i+1}}\) are homotopy equivalent to \(X_{t^{\prime }_i}\) and \(X_{t^{\prime }_{i+1}}\) respectively and compose the restrictions of \(r_i\) with these homotopy equivalences.
These maps determine homotopically \(f:X\rightarrow Y,\) when \(Y= \mathbb R \) or \(\mathbb{S ^1}\). For simplicity in writing, when \(Y= \mathbb R \) we put \(t_{k+1}\in (s_k, \infty )\) and \(t_1\in (\infty , s_1)\) and when \(Y=\mathbb{S }^1\) we put \(t_{k+1}=t_1 \in (s_k, s_1+2\pi ).\) All scalar or circlevalued simplicial maps on a simplicial complex, and all smooth maps with generic isolated critical points on a smooth manifold or stratified space are tame. In particular, Morse maps are tame.
2.1 Persistence and Invariants for RealValued Maps
Since our goal is to extend the notion of persistence from realvalued maps to circlevalued maps, we first summarize the questions that the persistence answers when applied to realvalued maps, and then develop a notion of persistence for circlevalued maps which can answer similar questions and more. We fix a field \(\kappa \) and write \(H_r(X)\) to denote the homology vector space of X in dimension r with coefficients in a field \(\kappa .\)
 Q1.
Does the class \(x\in H_r(X_{(\infty ,t]})\) originate in \(H_r(X_{(\infty ,t^{\prime \prime }]})\) for \(t^{\prime \prime }< t\)? Does the class \(x\in H_r(X_{(\infty ,t]})\) vanish in \(H_r(X_{(\infty ,t^{\prime }]})\) for \(t<t^{\prime }\)?
 Q2.
What are the smallest \(t^{\prime }\) and largest \(t^{\prime \prime }\) such that this happens?
 Q1.
Does the image of \(x\in H_r(X_t)\) vanish in \(H_r(X_{[t,t^{\prime }]}),\) where \(t^{\prime }>t\) or in \(H_r (X_{[t^{\prime \prime }, t]}),\) where \(t^{\prime \prime }<t\)?
 Q2.
Can x be detected in \(H_r(X_{t^{\prime }})\) where \(t^{\prime }>t\) or in \(H_r(X_{t^{\prime \prime }})\) where \( t^{\prime \prime } <t\)? The precise meaning of detection is explained below.
 Q3.
What are the smallest \(t^{\prime }\) and the largest \(t^{\prime \prime }\) for the answers to Q1 and Q2 to be affirmative?
We say that \(x\in H_r(X_t)\) is dead in \(H_r(X_{[t,t^{\prime }]}), \ t^{\prime }>t\), if its image by \(H_r(X_t)\rightarrow H_r(X_{[t, t^{\prime }]})\) vanishes. Similarly, x is dead in \(H_r(X_{[t^{\prime \prime },t]}), \ t^{\prime \prime }<t\), if its image by \(H_r(X_t)\rightarrow H_r(X _{[t^{\prime \prime }, t]})\) vanishes.
In Fig. 1, we indicate the bar codes both for sublevel and level persistence^{3} for some simple map \(f:X \rightarrow \mathbb R \) in order to illustrate their differences. The space X is a tube open on one end and f is the height function laid horizontally.
3 Persistence for CircleValued Maps
 Q4.
When \(x\in H_r(X_\theta )\) returns, how does the “returned class” compare with the original class x? It may disappear after going along the circle a number of times, or it might never disappear and if so how does this class change after its return.
 Bar codes: Intervals with ends \(s,s^{\prime }\) \(0<s\le 2\pi , \ s\le s^{\prime } < \infty \), that are closed or open at s or \(s^{\prime }\), precisely of one of the forms \([s,s^{\prime }], (s, s^{\prime }], [s,s^{\prime })\), and \((s,s^{\prime }).\) These intervals can be geometerized as “spirals” with equations in (1). For any interval \(\{s,s^{\prime }\}\) the spiral is the plane curve (see Fig. 3 in Sect. 4)$$\begin{aligned} \begin{array}{ccc} \begin{array}{c} x(\theta )= (\theta + 1s) \cos \theta \\ y(\theta )=(\theta +1s) \sin \theta \end{array}&\quad {\mathrm{{with}}}&\theta \in \{s, s^{\prime }\}. \end{array} \end{aligned}$$(1)
 Jordan cells. A Jordan cell is a pair \((\lambda , k),\) \(\lambda \in \overline{\kappa } \setminus 0, \ \ k \in \mathbb Z _{>0},\) where \(\overline{\kappa }\) denotes the algebraic closure of the field \(\kappa .\) It corresponds to a \(k\times k\) matrix of the form$$\begin{aligned} \left( \begin{array}{cccc} \lambda &{}\quad 1 &{}\quad 0\cdots &{}\quad 0\\ 0 &{}\quad \lambda &{}\quad 1\cdots &{}\quad 0\\ \vdots \\ 0 &{}\quad \cdots &{}\quad \lambda &{}\quad 1\\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad \lambda \end{array}\right) . \end{aligned}$$(2)

rInvariants. Given a tame map \(f: X\rightarrow \mathbb S ^1\), the collection of bar codes and Jordan cells for each dimension \(r\in \{ 0,1,2,\ldots , \dim X\}\) constitute the rinvariants of the map f.
The bar codes for f can be inferred from \(\tilde{f}: \tilde{X}_{[a,b]}\rightarrow \mathbb R \) with \([a,b]\) being any large enough interval. Specifically, the bar codes of \(f:X\rightarrow \mathbb{S }^1\) are among the ones of \(\tilde{f}: \tilde{X}_{[a,b]}\rightarrow \mathbb R \) for \((ba)\) being at most \(\sup _\theta \dim H_r(X_\theta ).\)
 1.
The Betti numbers of each fiber.
 2.
The Betti numbers of the source space X.
 3.
The dimension of the kernel and the image of the linear map induced in homology by the inclusion \(X_\theta \subset X\) as well as other additional topological invariants not discussed here [2].
Theorem 3.1
\(\dim H_r (X_\theta )=\sum _{B\in \mathcal B _r} n_\theta (B) + \sum _{J\in \mathcal J _r} n(J).\)
Theorem 3.2
\( \dim H_r (X)=\# \{B\in {\mathcal{B }}_r \mathrm{both\ ends\ closed}\} + \#\{B\in {\mathcal{B }}_{r1}  \mathrm{both\ ends} \mathrm{open}\} + \# \{J\in \mathcal{J }_{r} \lambda (J)=1\} +\# \{J\in \mathcal{J }_{r1} \lambda (J)=1\}. \)
Using the same arguments as in the proof of the above Theorems one can derive:
Proposition 3.3
\(\dim \mathrm{img}(H_r (X_\theta )\rightarrow H_r(X))= \# \{ B\in \mathcal{B }_r\,\, n_\theta (B)\ne 0\, \mathrm{and \ both} \mathrm{ends \ closed}\} + \# \{ J\in \mathcal{J }_{r} \lambda =1 \}\)
A realvalued tame map \(f: X\rightarrow \mathbb R \) can be regarded as a circlevalued tame map \(f^{\prime }:X\rightarrow \mathbb S ^1\) by identifying \(\mathbb R \) to \((0,2\pi )\) with critical values \(t_1,\ldots , t_m\) becoming the critical angles \(\theta _1,\ldots ,\theta _m\) where \(\theta _i=2\arctan t_i + \pi \). The map \(f^{\prime }\) in this case will not have any Jordan cells and the bar codes will be the same as level persistence bar codes. We have the following corollary:
Corollary 3.4
\( \dim H_r (X_\theta )=\sum _{B\in \mathcal{B }_r} n_\theta (B) \) and \(\dim H_r (X)=\# \{ B\in \mathcal{B }_r \mathrm{both\,ends} \mathrm{closed} \} + \# \{ B\in \mathcal{B }_{r1}  \mathrm{both\,ends\,open} \}. \)
Theorem 3.1 is quite intuitive and is in analogy with the derived results for sublevel and level persistence [4, 22]. Theorem 3.2 is more subtle. Its counterpart for realvalued function (Corollary 3.4) has not yet appeared in the literature though a related result for homology of source space can be derived from extended persistence [7]. The proofs of these results require the definition of the bar codes and Jordan cells which appear in the next section. The proofs are sketched in Sect. 5.
Questions Q1–Q3 can be answered using the bar codes. Question Q4 about returned homology can be answered using the bar codes and Jordan cells.
4 Representation Theory and its Invariants
The invariants for the circlevalued map are derived from the representation theory of quivers. The quivers are directed graphs. The representation theory of simple quivers such as paths with directed edges was described by Gabriel [15] and is at the heart of the derivation of the invariants for zigzag and then level persistence in [4]. For circlevalued maps, one needs representation theory for circle graphs with directed edges. This theory appears in the work of Nazarova [18], and Donovan and Freislich [11].
Let \(G_{2m}\) be a directed graph with \(2m\) vertices, \(x_1, x_1, \ldots , x_{2m}\). Its underlying undirected graph is a simple cycle. The directed edges in \(G_{2m}\) are of two types: forward \(a_i: x_{2i1}\rightarrow x_{2i}\), \(1\le i\le m\), and backward \(b_i: x_{2i+1}\rightarrow x_{2i}\), \(1\le i\le m1\), \(b_m:x_1\rightarrow x_{2m}\).
We think of this graph as being located on the unit circle centered at the origin o in the plane.
A representation \(\rho \) on \(G_{2m}\) is an assignment of a vector space \(V_x\) to each vertex x and a linear map \(\ell _e: V_{x}\rightarrow V_{y}\) for each oriented edge \(e=\{x,y\}\). Two representations \(\rho \) and \(\rho ^{\prime }\) are isomorphic if for each vertex x there exists an isomorphism from the vector space \(V_x\) of \(\rho \) to the vector space \(V^{\prime }_x\) of \(\rho ^{\prime }\), and these isomorphisms commute with the linear maps \(V_{x}\rightarrow V_{y}\) and \(V^{\prime }_{x}\rightarrow V^{\prime }_{y}\). A nontrivial representation assigns at least one vector space which is not zerodimensional. A representation is indecomposable if it is not isomorphic to the sum of two nontrivial representations.
Given two representations \(\rho \) and \(\rho ^{\prime }\), their sum \(\rho \oplus \rho ^{\prime }\) is a representation whose vector spaces are the direct sums \(V_x\oplus V^{\prime }_x\) related by linear maps that are the direct sums \(\ell _e\oplus \ell ^{\prime }_e.\) It is not hard to observe that each representation has a decomposition as a sum of indecomposable representations unique up to isomorphisms.

Bar code representation \(\rho ^I(\{i, j\};k)\): Suppose that the evenly indexed vertices \(\{x_2, x_4, \ldots , x_{2m}\}\) of \(G_{2m}\) which are the targets of the directed arrows correspond to the angles \(0< s_1< s_2 <\cdots <s_{m} \le 2\pi .\) Draw the spiral curve given by (1) for the interval \(\{s_{i}, s_{j}+2k\pi \}\); refer to Fig. 3. For each \(x_i\), let \(\{e_i^1,e_i^2,\ldots \}\) denote the ordered set (possibly empty) of intersection points of the ray \(ox_i\) with the spiral. While considering these intersections, it is important to realize that the point \((x(s_i), y(s_i))\) (resp. \((x(s_j+2k\pi ), y(s_j+2k\pi ))\)) does not belong to the spiral (1) if \(\{i,j\}\) is open at \(i\) (resp. \(j\)). For example, in Fig. 3, the last circle on the ray \(ox_{2j}\) is not on the spiral since \([i,j)\) in \(\rho ^I([i,j);2)\) is open at right. Let \(V_{x_i}\) denote the vector space generated by the base \(\{e_i^1,e_i^2,\ldots \}\). Furthermore, let \(\alpha _i: V_{x_{2i1}}\rightarrow V_{x_{2i}}\) and \(\beta _i: V_{x_{2i+1}}\rightarrow V_{x_{2i}}\) be the linear maps defined on bases and extended by linearity as follows: assign the vector \(e^h_{2i}\in V_{x_i}\) to \(e^{\ell }_{2i\pm 1}\) if \(e^h_{2i}\) is an adjacent intersection point to the points \(e^{\ell }_{2i\pm 1}\) on the spiral. If \(e^h_{2i}\) does not exist, assign zero to \(e^{\ell }_{2i\pm 1}\). If \(e^{\ell }_{2i\pm 1}\) does not go to zero, \(h\) has to be \(l\), \(l1\), or \(l+1\). The construction above provides a representation on \(G_{2m}\) which is indecomposable. Once the angles \(s_i\) are associated to the vertices \(x_{2i}\) one can also think of these representations \(\rho ^I(\{i,j\};k)\) as the bar codes \([s_{i}, s_{j}+2k\pi ]\), \((s_{i}, s_{j}+2k\pi ], [s_{i}, s_{j}+2k\pi )\), and \((s_{i}, s_{j}+2k\pi )\).

Jordan cell representation \(\rho ^J(\lambda ,k)\): Assign the vector space with the base \(\{e_1,e_2,\ldots ,e_k\}\) to each \(x_i\) and take all linear maps \(\alpha _i\) but one (say \(\alpha _1\)) and \(\beta _i\) the identity. The linear map \(\alpha _1\) is given by the Jordan matrix defined by \((\lambda ,k)\) in (2). Again this representation is indecomposable.
It follows from the work of [11, 18] that when \(\kappa \) is algebraically closed,^{5} the bar code and Jordan cell representations are all and only indecomposable representations of the quiver \(G_{2m}.\) The collection of all bar code and Jordan cell representations of a representation \(\rho \) constitutes its invariants.
Proposition 4.1
 1.
\(\dim (\rho _1\oplus \rho _2)= \dim \rho _1 + \dim \rho _2 \),
 2.
\(\dim \ker (\rho _1\oplus \rho _2)=\dim \ker \rho _1 + \dim \ker \rho _2\),
 3.
\(\dim \mathrm coker\ (\rho _1\oplus \rho _2)= \dim \mathrm coker\ \rho _1 + \dim \mathrm coker\ \rho _2\).
The description of a bar code representation permits explicit calculations.
Proposition 4.2
 1.If \(i\le j\) then
 (a)
\(\dim \rho ^I([i,j]; k)\) is given by: \(n_l=k+1\) if \((i+1)\le l\le j\) and k otherwise, \( r_l= k+1\) if \(i\le l\le j\) and k otherwise
 (b)
\(\dim \rho ^I((i,j]; k)\) is given by: \(n_l=k+1\) if \((i+1)\le l\le j\) and k otherwise, \(r_l= k+1\) if \((i +1)\le l\le j\) and k otherwise,
 (c)
\(\dim \rho ^I([i,j); k)\) is given by: \(n_l=k+1\) if \((i+1)\le l\le j\) and k otherwise, \( r_l= k+1\) if \(i\le l\le (j1)\) and k otherwise,
 (d)
\(\dim \rho ^I((i,j); k)\) is given by: \(n_l=k+1\) if \((i+1)\le l\le j\) and k otherwise, \(r_l= k+1\) if \((i +1)\le l\le ( j1)\) and k otherwise
 (a)
 2.If \(i> j\) then similar statements hold.
 (a)
\(\dim \rho ^I([i,j]; k)\) is given by: \(n_l=k\) if \((j+1)\le l\le i\) and \(k+1\) otherwise; \(r_l=k\) if \((j+1)\le l\le (i1)j\) and \(k+1\) otherwise
 (b)
\(\dim \rho ^I((i,j]; k)\) is given by: \(n_l=k\) if \((j+1)\le l\le i\) and \(k+1 \) otherwise. \(r_l=k\) if \((j+1)\le l\le i\) and \(k +1\) otherwise,
 (c)
\(\dim \rho ^I([i,j); k)\) is given by: \(n_l=k\) if \((j+1)\le l\le i\) and \(k+1\) otherwise; \(r_l=k\) if \( j\le l \le (i1)\) and \(k+1 \) otherwise,
 (d)
\(\dim \rho ^I((i,j); k)\) is given by: \(n_l=k\) if \((j+1)\le l\le i\) and \(k+1\) otherwise; \(r_l=k\) if \(j\le l\le i\) and \(k+1\) otherwise.
 (a)
 3.
\(\dim \rho ^J(\lambda ,k)\) is given by \(n_i=r_i=k\)
Proposition 4.3
 1.
\(\dim \ker \rho ^I([i,j]; k)=0\), \(\dim \mathrm coker\ \rho ^I([i,j]; k)=1\),
 2.
\(\dim \ker \rho ^I([i,j); k)=0\), \(\dim \mathrm coker\ \rho ^I([i,j); k)=0\),
 3.
\(\dim \ker \rho ^I ((i,j]; k)=0\), \(\dim \mathrm coker\ \rho ^I((i,j]; k)=0\),
 4.
\(\dim \ker \rho ^I((i,j); k)=1\), \(\dim \mathrm coker\ \rho ^I((i,j); k)=0\),
 5.
\(\dim \ker \rho ^{J}(\lambda ,k) =0\) (resp. 1) if \(\lambda {\ne 1}\) (resp. 1),
 6.
\(\dim \mathrm coker\ \rho ^{J}(\lambda ,k)=0\) (resp. 1) if \(\lambda {\ne 1}\) (resp. 1).
Observation 4.4
Definition 4.5
(\(r\)Invariants.) Let f be a circlevalued tame map defined on a topological space X. For f with m critical angles \(0< s_1< s_2,\ldots , s_m\le 2\pi \), consider the quiver \(G_{2m}\) with the vertices \(x_{2i}\) identified with the angles \( s_i\) and the vertices \(x_{2i1}\) identified with the angles \(t_i\) that satisfy \(0<t_1<s_1<t_2 < s_2, \ldots , t_m<s_m.\)
For any \(r\), consider the representation \(\rho _r\) of \(G_{2m}\) with \(V_{x_i}=H_r(X_{x_i})\) and the linear maps \(\alpha _i\)s and \(\beta _i\)s induced in the \(r\)homology by maps \(a_i: X_{x_{2i1}} \rightarrow X_{x_{2i}}\) and \(b_i: X_{x_{2i+1}} \rightarrow X_{x_{2i}}\) described in Sect. 2. The bar code and Jordan cell representations of \(\rho _r\) are independent of the choice of \(t_i\)s and are collectively referred as the rinvariants of \(f.\)
5 Proof of the Main Results
Figure 2 and the bar codes listed below suggest why a semiclosed (one end open and the other closed) bar code does not contribute to the homology of the total space X and why a closed bar code (both ends closed) in \(\mathcal{B}_r\) contributes one unit while an open (both ends open) bar code in \(\mathcal{B}_{r1}\) contributes one unit to the \(H_r(X)\). Indeed, in our example, a semiclosed bar code in \(\mathcal{B}_1\) adds to the total space a cone over \(\mathbb{S }^1,\) which is a contractible space. It gets glued to the total space along a generator of the cone (a segment connecting the apex to \(\mathbb{S }^1\)), again a contractible space. A closed bar code in \(\mathcal{B}_1\) adds a cylinder of \(\mathbb{S }^1\) whose \(H_1\) has dimension \(1\). It gets glued to the total space along a generator of the cylinder (a segment connecting the same point on the two copies of \(\mathbb{S }^1\)), again a contractible space. An open bar code in \(\mathcal{B}_1\) adds the suspension over \(\mathbb{S }^1\), topologically a \(2\)sphere which gets glued along a meridian, a contractible space. This contributes a dimension to \(H_2\).
The lack of contribution of a Jordan cell with \(\lambda \ne 1\) as well as the contribution of one unit of a Jordan cell in \(\mathcal{J}_r\) with \(\lambda =1\) to both \(r\) and \(r+1\) dimensional homology of the total space should not be a surprise for the reader familiar with the calculation of the homology of mapping torus.
Below we explain rigorously but schematically the arguments for the proof of Theorems 3.1, 3.2, and Corollary 3.4.
The proof of Theorem 3.1 is a consequence of Propositions 4.1 and 4.2. The proof of Theorem 3.2 proceeds along the following lines.
 1.
\(\mathcal T = \mathcal P ^{\prime } \cup \mathcal P ^{\prime \prime }\),
 2.
\(\mathcal P ^{\prime } \cap \mathcal P ^{\prime \prime }= (\bigsqcup _{1\le i\le m} R_i\times (\varepsilon ,1\varepsilon ))\sqcup \mathcal X \), and
 3.
the inclusions \((\bigsqcup _{1\le i\le m} R_i\times \{ 1/2 \})\sqcup \mathcal X \subset \mathcal P ^{\prime }\cap \mathcal P ^{\prime \prime }\) as well as the obvious inclusions \(\mathcal X \subset \mathcal P ^{\prime }\mathrm{and} \mathcal X \subset \mathcal P ^{\prime \prime }\) are homotopy equivalences.
6 Algorithm
Given a circlevalued tame map \(f:X\rightarrow \mathbb S ^1\), we now present an algorithm to compute the bar codes and the Jordan cells when X is a finite simplicial complex, and f is generic and linear. This makes the map tame. Genericity means that f is injective on vertices. To explain linearity we recall that, for any simplex \(\sigma \in X\), the restriction \(f_{\sigma }\) admits liftings \(\hat{f}:\sigma \rightarrow \mathbb R \), i.e. \(\hat{f}\) is a continuous map which satisfies \(p\cdot \hat{f}= f_\sigma .\) The map \(f: X\rightarrow \mathbb S ^1\) is called linear if for any simplex \(\sigma \), at least one of the liftings (and then any other) is linear.
Our algorithm takes the simplicial complex X equipped with the map f as input and, for any \(r\), computes the matrix \(M_{\rho _r}\) of the representation \(\rho _r\) for f. This requires recognizing the critical values \(s_1, s_2, \ldots , s_{m}\in \mathbb S ^1\) of f, and for conveniently chosen regular values \(t_1, t_2, \ldots , t_{m}\in \mathbb S ^1\), determining the vector spaces \(V_{2i1}=H_r(X_{t_i}), V_{2i}=H_r(X_{s_i})\) with the linear maps \(\alpha _i\) and \(\beta _i\) as matrices. We consider the block matrix \(M_{\rho _r}: \bigoplus _{1\le i\le m} V_{2i1} \rightarrow \bigoplus _{1\le i\le m}V_{2i}\) described in the previous section.

Step 1. Compute the matrices \(\alpha _i\), \(\beta _i\) that constitute the matrix \(M_{\rho _r}\) of the representation \(\rho _r\).

Step 2. Process the matrix of \(M_{\rho _r}\) to derive the bar codes ending up with a representation \(\rho ^{\prime }_r\) whose all \(\alpha _i^{\prime }\)s and \(\beta _i^{\prime }\)s are invertible matrices.

Step 3. Compute the Jordan cells of \(\rho _r\) from the representation \(\rho _r^{\prime }.\)
 1.
\(\sigma (i):=\sigma \cap X_{t_i}\) with \(\dim (\sigma (i))= \dim \sigma 1\) if the intersection is nonempty
 2.
\(\sigma \langle i\rangle := \sigma \cap X_{(t_i, t_{i+1})}\) with \(\dim \sigma \langle i \rangle = \dim \sigma \) if the intersection is nonempty.
The incidence matrix of \(A=X_i\sqcup X_{i+1}\) appears in the upper left corner of the matrix for \(R:=R_i\). We obtain the matrices for the linear maps \(\alpha _i:H_r(X_{t_{i}})\rightarrow H_r(X_{s_i})\) and \(\beta _i: H_r(X_{t_{i+1}})\rightarrow H_r(X_{s_{i}})\) by using the persistence algorithm [5, 22] on \(R\) and \(A\) as follows. First, we run the persistence algorithm on the incidence matrix for \(A\) to compute a base of the homology group \(H_r(A)\). We continue the procedure by adding the columns and rows of the matrix for \(R\) to obtain a base of \(H_r(R)\). It is straightforward to compute a matrix representation of the inclusion induced linear map \(H_r(A)\rightarrow H_r(R)\) with respect to the bases computed by the persistence algorithm.
Step 2. Step 2 takes the matrix representation \(M_{\rho _r}\) constructed out of matrices \(\alpha _i, \beta _i\) computed in step 1, and uses four elementary transformations \(T_1(i),T_2(i),T_3(i)\), and \(T_4(i)\) defined below to transform \(M_{\rho _r}\) to \(M_{\rho ^{\prime }_{r}}= T_{\cdots } (\cdots ) M_{\rho _r},\) whose total number of rows and columns is strictly smaller than that of \(M_{\rho _r}\). For convenience, let us write \(\rho =\rho _r\) and \(\rho ^{\prime }=\rho _r^{\prime }\). Each elementary transformation \(T\) modifies the representation \(\rho \) to the representation \(\rho ^{\prime }\) while keeping indecomposable Jordan cell representations unaffected but possibly changing the bar code representations. Some of these bar code representations remain the same, some are eliminated, and some are shortened by one unit as described below. For each elementary transformation we record the changes to reconstruct the original bar codes. The elementary transformations are applied as long as the linear maps \(\alpha _i\) or \(\beta _i\) satisfy some injectivity and surjectivity property. When no such transformation is applicable, the algorithm terminates with all \(\alpha _i\) and \(\beta _i\) being necessarily invertible matrices. At this point the bar codes can be reconstructed reading backwards the eliminations/modifications performed. The Jordan cells then can be obtained as detailed in Step 3.

\(T_1(i)\) shortens the bar codes\((i1, k\}\) to \((i,k\}\) if \(i\ge 2\) and shortens the bar codes \((m, k\}, m<k\), to \((1, km\}\) if \(i=1\).

\(T_2(i)\) shortens the bar codes \(\{l, i+km]\) to \(\{l, i1+km]\) for \(k\ge 0.\)

\(T_3(i)\) shortens the bar codes \([i, k\}\) to \([i+1,k\}\) for \(i< m\) and to \([1, km\}\) if \(i=m\).

\(T_4(i)\) shortens the bar codes \(\{l, (i+1)+km )\) to \(\{l, i+km)\) for \(k\ge 0.\)
To decide how many bar codes are eliminated one uses Proposition 6.1 below. Let \(\#\{i,j\}_\rho \) denote the number of bar codes of type \(\{i, j\}.\)
Proposition 6.1
 1.
\(\# (i,i+1)_\rho = \dim (\ker \beta _i\cap \ker \alpha _{i+1})\)
 2.
\(\# [i,i]_\rho = \dim (V_{2i}/((\beta _i (V_{2i+1}) + \alpha _i (V_{2i1}))\)
 3.
\(\# (i,i+1]_\rho = \dim (\beta _i(V_{2i+1}) +\alpha _i(\ker \beta _{i1}))  \dim (\beta _i(V_{2i+1}))\)
 4.
\(\# [i, i+1)_\rho = \dim (\alpha _i(V_{2i1}) +\beta _i(\ker \alpha _{i+1}))\dim (\alpha _i(V_{2i1}))\)
 Transformation \(T_1(i)\): \(V^{\prime }_{2i1}= V_{2i1}/ \ker \beta _{i1}, \quad V^{\prime }_{2i}= V_{2i}/ \alpha _{i}(\ker \beta _{i1} ),\quad V^{\prime }_k= V_k, k= 2i, 2i1\)
 Transformation \(T_2(i)\): \(V^{\prime }_{2i}= \beta _i(V_{2i+1}), \quad V^{\prime }_{2i1}= \alpha ^{1}_{i}(\beta _i(V_{2i+1})), \quad V^{\prime }_k= V_k, k\ne 2i1, 2i\)
 Transformation \(T_3(i)\): \(V^{\prime }_{2i}= \alpha _{i}(V_{2i1}), \quad V^{\prime }_{2i+1}= \beta _i^{1}(\alpha _i(V_{2i1})), \quad V^{\prime }_k= V_k, k\ne 2i, 2i+1\)
 Transformation \(T_4(i)\): \(V^{\prime }_{2i+1}= V_{2i+1}/ \ker \alpha _{i+1}, \quad V^{\prime }_{2i}= V_{2i}/ \beta _i(\ker \alpha _{i+1} ), \quad V^{\prime }_k= V_k, k\ne 2i, 2i+1.\)
As one can see from the diagrams above, when \(\beta _{i1}\) is injective, the representations \(\rho \) and \(\rho ^{\prime }\) are the same and we say that \(T_1(i)\) is not applicable. Similarly, when \(\beta _{i}\) is surjective, \(T_2(i)\) is not applicable, when \(\alpha _{i}\) is surjective, \(T_3(i)\) is not applicable, and when \(\alpha _{i+1}\) is injective, \(T_4(i)\) is not applicable. When all \(\alpha _i, \beta _i\) are invertible, no elementary transformation is applicable and at this stage the algorithm (Step 2) terminates.
 1.If \(\beta _{i1}\) is not injective, we apply \(T_1(i).\) This boils down to changing the bases of \(V_{2i1}\) and \(V_{2i}\) so that the matrix \(B_{2(i1)}\) becomes with (\(\beta _{i1,1} \, 0\)) in column echelon form andin row echelon form. In this block matrix the first and third columns correspond to \(V^{\prime }_{2i1}\) and \(V_{2i+1}\) respectively, and the first and third rows to \(V_{2(i1)}\) and \(V^{\prime }_{2i}\) respectively. The second column and row become “irrelevant” as a result of which the modified block matrix \(B_{2(i1)}\) becomes$$\begin{aligned} \left( \begin{array}{c}\alpha ^1_{i,2}\\ 0\end{array}\right) \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\beta ^{\prime }_{i1}&{}\quad 0\\ \alpha ^{\prime }_i&{}\beta ^{\prime }_i\end{array}\right) = \left( \begin{array}{cc}\beta _{i1,1}&{}\quad 0\\ \alpha ^2_{i,1}&{} \quad \beta ^2_{i} \end{array}\right) . \end{aligned}$$
 2.If \(\beta _{i}\) is not surjective, we apply \(T_2(i).\) This boils down to changing the bases of \(V_{2i1}\) and \(V_{2i}\) so that the matrix \(B_{2(i1)}\) becomes within row echelon form and (\(\alpha ^2_{i,1}\, 0\)) in column echelon form. In this block matrix the second and third columns correspond to \(V^{\prime }_{2i1}\) and \(V_{2i+1}\) respectively, and the first and second rows to \(V_{2(i1)}\) and \(V^{\prime }_{2i}\) respectively. We make the first column and third row “irrelevant” as a result of which the modified block matrix \(B_{2(i1)}\) becomes$$\begin{aligned} \left( \begin{array}{c}\beta ^1_{i}\\ 0 \end{array}\right) \end{aligned}$$When \(B_{2i1}\) is processed then:$$\begin{aligned} \left( \begin{array}{cc}\beta ^{\prime }_{i1}&{}\quad 0\\ \alpha ^{\prime }_i&{}\quad \beta ^{\prime }_i\end{array}\right) = \left( \begin{array}{cc}\beta _{i1,2}&{}\quad 0\\ \alpha ^1_{i,2}&{} \quad \beta ^1_{i}\end{array}\right) . \end{aligned}$$
 3.If \(\alpha _i\) is not surjective, we apply \(T_3(i).\) This boils down to changing the bases of \(V_{2i+1}\) and \(V_{2i}\) so that the matrix \(B_{2i1}\) becomes within row echelon form and \(\beta ^2_{i,1}\, 0\) in column echelon form. In this block matrix the first and third columns correspond to \(V_{2i1}\) and \(V^{\prime }_{2i+1}\) respectively, and the first and third rows to \(V^{\prime }_{2i}\) and \(V_{2i+2}\) respectively. We make the second column and second row “irrelevant” as a result of which the modified block matrix \(B_{2i1}\) becomes$$\begin{aligned} \left( \begin{array}{c}\alpha ^1_{i}\\ 0\end{array}\right) \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\alpha ^{\prime }_{i}&{}\quad \beta ^{\prime }_i\\ 0&{}\quad \alpha ^{\prime }_{i+1}\end{array}\right) = \left( \begin{array}{cc}\alpha ^1_{i}&{}\quad \beta ^1_{i,2}\\ 0 &{}\quad \alpha _{i+1,2}\end{array}\right) . \end{aligned}$$
 4.If \(\alpha _{i+1}\) is not injective, we apply \(T_4(i).\) This boils down to changing the bases of \(V_{2i+1}\) and \(V_{2i}\) so that the matrix \(B_{2i1}\) becomes with (\(\alpha _{i+1,1}\,0 \)) in column echelon form andin row echelon form. In this block matrix first and second columns correspond to \(V_{2i1}\) and \(\ V^{\prime }_{2i+1}\) respectively, and second and third rows to \(V_{2i}^{\prime }\) and \(V_{2(i+1)}\) respectively. We make the third column and first row “irrelevant” as a result of which the modified block matrix \(B_{2i1}\) becomes$$\begin{aligned} \left( \begin{array}{c}\beta ^1_{i,2}\\ 0\end{array}\right) \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\alpha ^{\prime }_{i}&{}\quad \beta ^{\prime }_i\\ 0&{}\quad \alpha ^{\prime }_{i+1}\end{array}\right) = \left( \begin{array}{cc}\alpha ^2_{i}&{}\quad \beta ^2_{i,1}\\ 0&{}\quad \alpha _{i+1,1}\end{array}\right) . \end{aligned}$$
When a bar code \([i,i]\) is eliminated, say, in the kth pass, we know that it corresponds to a bar code \([ik+1, i+k1]\) in the original representation. Similarly, other bar codes of type \(\{i,i+1\}\) eliminated at the kth pass correspond to the bar code \(\{ik+1,i+k\}\). In both cases, the multiplicity of the bar codes can be determined from the multiplicity of the eliminated bar codes thanks to Proposition 6.1.
When \(m=1\), the operations on above minors are not well defined. In this case we extend the quiver \(G_2\) to \(G_4\) (\(m=2\)) by adding fake levels \(t_2, s_2\) where \(H_r(X_{t_2})=H_r(X_{s_2})=H_r(X_{s_1})\) and \(\alpha _2,\beta _2\) are identities.^{7}
Example.

Inspect \(B_1\) and \(B_2.\) No changes are necessary.
 Inspect \(B_3\). Since \(\alpha _3\) is not injective, one modifies the block by applying \(T_4(2)\) which makes both \(\alpha _3\) and \(\beta _2\) equal to$$\begin{aligned} \left( \begin{array}{cc}1 &{}\quad 0\\ 0 &{}\quad 1\end{array}\right) . \end{aligned}$$

Inspect the blocks \(B_4, B_5, B_6, B_7.\) No changes are necessary.
 Inspect \(B_8\). Since \(\beta _4\) is not injective, one modifies the block by applying \(T_1(1)\) which leads to$$\begin{aligned} \alpha _1=\left( \begin{array}{cc}4 &{}\quad 3\\ 3 &{}\quad 0 \end{array}\right) \quad {\mathrm{{and}}}\quad \beta _1=\left( \begin{array}{cc}1 &{}\quad 1\\ 1 &{}\quad 0\end{array}\right) . \end{aligned}$$
Since \(\beta _4\) is already in column echelon form one only has to change the base of \(V_2\) to bring the last column of \(\alpha _1\) in row echelon form which ends up with
6.1 Implementation of \(T_1(i), T_2(i), T_3(i)\) and \(T_4(i).\)
 1.\(T_1 (i)\) acts on the block matrixFirst we modify \(B_{2(i1)}\) to the block matrix$$\begin{aligned} B_{2(i1)}=\left( \begin{array}{cc}\beta _{i1}&{}\quad 0\\ \alpha _i&{}\quad \beta _i \end{array}\right) . \end{aligned}$$where \(\begin{array}{cc}\beta _{i1,1}&\quad 0 \end{array}= \beta _{i1}\cdot R(\beta _{i1}) \quad {\mathrm{{and}}}\,\,\begin{array}{cc} \alpha _{i,1}&\quad \alpha _{i,2} \end{array}= \alpha _i\cdot R(\beta _{i1}).\) Recall the definition of \(R(\cdot )\) and \(L(\cdot )\) given under notations in the introduction. Then, one passes to the block matrix$$\begin{aligned} \left( \begin{array}{ccc}\beta _{i1,1}&{}\quad 0&{}\quad 0\\ \alpha _{i,2}&{}\quad \alpha _{i,2}&{} \quad \beta _i \end{array}\right) \end{aligned}$$with$$\begin{aligned} \left( \begin{array}{ccc} \beta _{i1,1}&{}\quad 0&{}\quad 0\\ \alpha ^1_{i,2}&{}\quad \alpha ^1_{i,2}&{}\quad \beta ^1_i\\ \alpha ^2_{i,2}&{}\quad 0&{}\quad \beta ^2_i\end{array}\right) \end{aligned}$$The modified block matrix is$$\begin{aligned} \left( \begin{array}{c}\alpha ^1_{i,2}\\ 0 \end{array}\right) =L(\alpha _{i,2})\cdot \alpha _{i,2}, \left( \begin{array}{c}\alpha ^1_{i,1}\\ \alpha ^2_{i,2} \end{array}\right) =L(\alpha _{i,2}) \cdot \alpha _{i,1} \quad {\mathrm{{and}}}\quad \left( \begin{array}{c}\beta ^1_{i}\\ \beta ^2_i \end{array}\right) =L(\alpha _{i,2})\beta _{i}. \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\beta _{i1,1}&{}\quad 0\\ \alpha ^2_{i,1}&{}\quad \beta ^2_i \end{array}\right) . \end{aligned}$$
 2.\(T_2 (i)\) acts on the block matrixFirst we modify \(B_{2(i1)}\) to the block matrix$$\begin{aligned} B_{2(i1)}= \left( \begin{array}{cc}\beta _{i1}&{}\quad 0\\ \alpha _i&{}\quad \beta _i \end{array}\right) . \end{aligned}$$where$$\begin{aligned} \left( \begin{array}{cc} \beta _{i1}&{}\quad 0\\ \alpha ^1_{i}&{}\quad \beta ^1_i\\ \alpha ^2_i&{}\quad 0 \end{array}\right) \end{aligned}$$Then, one passes to the block matrix$$\begin{aligned} \left( \begin{array}{c}\beta ^1_{i}\\ 0\end{array}\right) =L(\beta _i)\cdot \beta _i \quad {\mathrm{{and}}} \,\, \left( \begin{array}{c}\alpha ^1_{i}\\ \alpha ^2_i\end{array}\right) =L(\beta _i)\cdot \alpha _i. \end{aligned}$$with$$\begin{aligned} \left( \begin{array}{ccc}\beta _{i1,1}&{}\quad \beta _{i1,2}&{}\quad 0\\ \alpha ^1_{i,1}&{}\quad \alpha ^1_{i,2}&{}\quad \beta ^1_i\\ \alpha ^2_{i,1}&{}\quad 0&{}\quad 0\end{array}\right) \end{aligned}$$The modified block matrix is$$\begin{aligned} \left( \begin{array}{c}\alpha ^2_{i,1}\\ 0 \end{array}\right) \!=\! \alpha ^2_{i,1}\cdot R(\alpha ^2_{i,1}), \left( \begin{array}{c}\alpha ^1_{i,1}\\ \alpha ^1_{i,2} \end{array}\right) \!=\! \alpha _{i,1}R(\alpha ^2_{i,1}), \quad {\mathrm{{and}}} \,\, \left( \begin{array}{c} \beta _{i1,1}\\ \beta _{i1,2} \end{array}\right) = \beta _{i1}R(\alpha _{i,1}). \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\beta _{i1,2}&{}\quad 0\\ \alpha ^1_{i,2} &{}\quad \beta ^1_i \end{array}\right) . \end{aligned}$$
 3.\(T_3 (i)\) acts on the block matrixFirst we modify \(B_{2i1}\) to the block matrix$$\begin{aligned} B_{2i1}= \left( \begin{array}{cc} \alpha _{i}&{}\quad \beta _i\\ 0 &{}\quad \alpha _{i+1} \end{array}\right) . \end{aligned}$$where$$\begin{aligned} \left( \begin{array}{cc} \alpha ^1_i&{}\quad \beta ^1_i\\ 0 &{}\quad \beta ^2_i\\ 0&{}\quad \alpha _{i+1} \end{array}\right) \end{aligned}$$Then, one passes to the block matrix$$\begin{aligned} \left( \begin{array}{c}\alpha ^1_{i}\\ 0\end{array}\right) =\alpha _i\cdot R(\alpha _i) \,\, {\mathrm{{and}}} \left( \begin{array}{c}\beta ^1_{i}\\ \beta ^2_i\end{array}\right) =\beta _i\cdot R(\alpha _i). \end{aligned}$$with$$\begin{aligned} \left( \begin{array}{ccc} \alpha ^1_i&{}\quad \beta ^1_{i,1}&{}\quad \beta ^1_{i,2}\\ 0 &{}\quad \beta ^2_{i,1}&{}\quad 0\\ 0&{}\quad \alpha _{i+1,1}&{}\quad \alpha _{i+1,2}\end{array}\right) \end{aligned}$$and \(\begin{pmatrix}\alpha _{i+1,1}&\alpha _{i+1,2} \end{pmatrix} = \alpha _{i+1}\cdot R( \beta ^2_{i}).\) The modified block matrix is$$\begin{aligned} \left( \begin{array}{cc}\beta ^2_{i,1}&\quad 0\end{array}\right) = \beta ^2_{i}\cdot R( \beta ^2_{i}), \left( \begin{array}{cc}\beta ^1_{i,1}&\quad \beta ^1_{i,2} \end{array}\right) = \beta ^1_i\cdot R( \beta ^2_{i}) \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\alpha ^1_{i}&{}\quad \beta ^1_{i,2}\\ 0 &{}\quad \alpha _{i+1,2}\end{array}\right) . \end{aligned}$$
 4.\(T_4 (i)\) acts on the block matrixFirst one modifies \(B_{2i1}\) to the block matrix$$\begin{aligned} B_{2i1}=\left( \begin{array}{cc} \alpha _{i}&{}\quad \beta _i\\ 0 &{}\quad \alpha _{i_1} \end{array}\right) . \end{aligned}$$where$$\begin{aligned} \left( \begin{array}{ccc} \alpha _i&{}\quad \beta _{i,1}&{}\quad \beta _{i,2} \\ 0&{}\quad \alpha _{i+1,1}&{}\quad 0 \end{array}\right) \end{aligned}$$Then, one passes to the block matrix$$\begin{aligned} \left( \begin{array}{cc}\alpha _{i+1,1}&\quad 0\end{array}\right) =\alpha _{i+1}\cdot R(\alpha _{i+1}) \quad {\mathrm{{and}}}\,\, \left( \begin{array}{cc}\beta _{i,1}&\quad \beta _{i,2} \end{array}\right) =\beta _i\cdot R(\alpha _{i+1}). \end{aligned}$$with$$\begin{aligned} \left( \begin{array}{ccc} \alpha ^1_i&{}\quad \beta ^1_{i,1}&{}\quad \beta ^1_{i,2} \\ \alpha ^2_i&{} \beta ^2_{i,1}&{}\quad 0\\ 0 &{}\quad \alpha ^2_{i+1,1}&{}\quad 0\end{array}\right) \end{aligned}$$The modified block matrix is$$\begin{aligned} \left( \begin{array}{c}\beta ^1_{i,2}\\ 0 \end{array}\right) = L(\beta _{i,2})\cdot \beta _{i,2}, \left( \begin{array}{c}\beta ^1_{i,1}\\ \beta ^2_{i,1} \end{array}\right) =L(\beta _{i,2})\cdot \beta _{i,1} \quad {\mathrm{{and }}}\,\, \left( \begin{array}{c}\alpha ^1_{i}\\ \alpha ^2_{i} \end{array}\right) = L(\beta _{i,2})\cdot \alpha _{i}. \end{aligned}$$$$\begin{aligned} \left( \begin{array}{cc}\alpha ^2_i&{}\quad \beta ^2_{i,1}\\ 0&{}\quad \alpha _{i+1,1}\end{array}\right) . \end{aligned}$$
6.2 Time Complexity
Let the input complex X have n simplices in total on which the circlevalued map f is defined which has m critical values.
Then, step 1 takes \(O(nd)\) time to detect all the critical values where \(d\le n\) is the maximum degree of any vertex. The critical values can be computed by looking at the simplices adjacent to each of the vertices. To compute the matrices \(\alpha _i\) and \(\beta _i\), we set up the matrices of size \(O(n)\times O(n)\) and run persistence on them. Using the algorithm of [17], this can be achieved in \(O(M(n))\) time where \(M(n)\) is the time complexity of multiplying two \(n\times n\) matrices.^{8} Since we perform this operations for each of the critical levels and the spaces between them, we have \(O(m M(n))\) total time complexity for step 1.
In step 2, we process the matrix \(M_{\rho _r}\) iteratively until all bar code representations are removed. In each pass except the last one, we are guaranteed to shrink a bar code by at least one unit. Therefore, the total number of passes is bounded from above by the total length of all bar codes. Theorem 3.1 implies that a bar code cannot come back to the same level more than \(\max _{s_i} \dim H_r(X_{s_i})\) times which can be at most \(O(n)\). Therefore, any bar code has a length of at most \(O(nm)\) giving a total length of \(O(n^2m)\) over all bar codes. Hence, the repeat loop in the algorithm BarCode cannot have more that \(O(n^2m)\) iterations. In each iteration, we reduce the block matrices each of which can be done with \(O(M(n))\) matrix multiplication time [16]. Since there are at most \(O(m)\) block matrices to be considered, we have \(O(m M(n))\) time per iteration giving a total of \(O(n^2m^2 M(n))\) time for step 2.
Step 3 is performed on the resulting matrix from step 2 which has \(O(mn)\times O(mn)\) size. This can again be performed by matrix multiplication which takes \(O(M(mn))\) time.
Therefore, the entire algorithm has time complexity of \(O(m^2n^2 M(n)+ M(mn))\).
7 Conclusions
We have analyzed circlevalued maps from the perspective of topological persistence. We show that the notion of persistence for such maps incorporate an invariant that is not encountered in persistence studied erstwhile. Our results also shed lights on computing homology vector spaces and other topological invariants from bar codes and Jordan cells (Theorems 3.1 and 3.2). We have given an algorithm to compute the bar codes and the Jordan cells; the algorithms can also be adapted to compute zigzag persistence. In a subsequent work, Burghelea and Haller have derived more subtle topological invariants like Novikov homology, monodromy [2], Reidemeister torsion, and others from bar codes and Jordan cells confirming their mathematical relevance. We have not treated in this paper the stability of the invariants; see [2] for partial answer.
The standard persistence is related to Morse theory. In a similar vein, the persistence for circlevalued map is related to Morse Novikov theory [19]. The work of Burghelea and Haller applies Morse Novikov theory to instantons and closed trajectories for vector field with Lyapunov closed one form [1]. The results in this paper will very likely provide additional insight on the dynamics of these vector fields and have implications in computational topology in particular and algebraic topology in general.
Footnotes
 1.
It was brought to our attention by David CohenSteiner that the extended persistence proposed in [7] allows similar connections between homology of source spaces and persistence.
 2.
Since the map f is proper and \(\mathbb S ^1\) compact, so is X.
 3.
The white circles indicate open ends and the dark circles indicate closed ends.
 4.Each circle is oriented counterclockwise and represents a 1dimensional homology class; “k times (\(k\) times) around the circle” means “going around k times counter clockwise (clockwise respectively)”.
 5.
When \(\kappa \) is not algebraically closed Jordan cells have to be replaced by conjugate classes of indecomposable (not conjugated to a direct sum of matrices) matrices with entries in \(\kappa \).
 6.
If \(i=1\) eliminates the bar codes \((m,m+1)\) and \((m,m+1]\).
 7.
Other easier methods can also be used in this case.
 8.
We have \(M(n)=O(n^{\omega })\) where \(\omega < 2.376\) [8].
Notes
Acknowledgments
We acknowledge the support of the NSF Grant CCF0915996 which made this research possible. We also thank all the referees whose comments were helpful in improving the presentation of the paper.
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