Fiber of persistent homology on morse functions

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INTRODUCTION
Persistent Homology is a central and computable descriptor in Topological Data Analysis (TDA) which has been applied to a large variety of data science problems.Namely the persistence map PH associates to a realvalued function f on a topological space X a so-called barcode that captures the topological variations of its sub level-sets (Edelsbrunner and Harer, 2008;Zomorodian and Carlsson, 2005).It is natural to ask how much information can be recovered from persistent homology: Given a barcode D what can we say about the fiber PH −1 (D)?
For the purpose of this paper X = M is an arbitrary smooth (finite-dimensional) compact manifold with boundary ∂M and f : M → R is a Morse function, i.e. f has a prescribed constant value a j on each boundary component ∂M j and has isolated critical points none of which belong to ∂M .Denote by D := PH(f ) the associated barcode.Acting on f by pre-composition with the group D Id (M ) of diffeomorphisms which are isotopic to the identity, we get an orbit O Id (f ) inside the space of Morse functions.Our core contribution (Theorem 3.1) is the equality between this orbit and the the path connected component PH −1 f (D) containing f in the fiber of PH over D: PH −1 f (D) = O Id (f ).This result crucially relies on Mather's fibration theorem for smooth mappings (Mather, 1969), which we slightly adapt to the case of Morse functions with equal critical values using results of Cerf (1970).
We then put at work the abundant literature about the homotopy type of the orbit O Id (f ), especially the work of Maksymenko (2006): the mapping φ → f • φ in fact defines a locally trivial fibration from D Id (M ) to the orbit O Id (f ), with fiber S Id (f ) ⊆ D Id (M ) the diffeomorphisms stabilising f , i.e. f • φ = f .Hence a long exact sequence links the fiber PH −1 f (D) = O Id (f ) to well-studied diffeomorphism groups of the manifold M .In particular for any compact surface M , we compute π n (PH −1 f (D)) for n ≥ 2 (Proposition 4.6).In fact, if D has distinct interval endpoints we derive the complete homotopy type of PH −1 f (D) for many compact surfaces (Propositions 4.8 and 4.9).
Variations of this setting have already been addressed.In the discrete setting where X = K is a finite simplicial complex and f is compatible with face inclusions, the fiber PH −1 (D) is a complex of polyhedra (Leygonie and Tillmann, 2022).This structure can be used to algorithmically reconstruct the fiber (Leygonie and Henselman-Petrusek, 2021).In the restricted case where K is a line complex, each path connected component of PH −1 (D) is contractible (Cyranka et al., 2020), and it is homeomorphic to a circle in the case where K is a subdivision of the unit circle (Mischaikow and Weibel, 2021).
In the analogous, continuous 1-dimensional setting where X is the interval or the circle and f is continuous, each component in the fiber is contractible and circular respectively as we show in Section 5.For the unit interval it is possible to count the number of path connected components in PH −1 (D) by means of the combinatorics of the barcode (Curry, 2018).For higher dimensional X analyzing the fiber is a challenging problem: already for Morse functions on the 2-sphere X = S 2 new tools have been designed to describe the fiber PH −1 (D), and allowed for conjectures on the number of path connected components (Catanzaro et al., 2020).However the higher dimensional homotopy groups of PH −1 (D) remain unknown, from which stems the motivation of this work.Acknowledgements.We wish to thank Ulrike Tillmann for her various insights on this project.

STABILITY OF MORSE FUNCTIONS
We fix a d-dimensional compact smooth manifold M with boundary ∂M , whose path connected components are denoted by ∂M j .Given another smooth manifold N , we denote by C ∞ (M, N ) the space of smooth maps from M to N equipped with the C ∞ Whitney topology.We denote by D(M ) ⊆ C ∞ (M, M ) the diffeomorphisms of M , and by D Id (M ) its subspace of Id-isotopic diffeomorphisms, i.e. the path connected component of the identity map.
Given real values a j , a smooth map f belongs to the space Morse(M ) ⊆ C ∞ (M, R) of Morse functions on M if: • The Hessian of f is non-degenerate at critical points, all of which belong to M \ ∂M ; and • The restrictions f |∂Mj to each boundary component ∂M j are constant with prescribed value a j .Then D(M ) acts on C ∞ (M, R) by pre-composition and we denote by O(f ) ⊆ Morse(M ) the orbit of f , and by O Id (f ) ⊆ O(f ) the orbit of f under the restricted action of Id-isotopic diffeomorphisms.
To express the local triviality results of this section, we rely on the notion of local cross-sections defined below.
Definition 2.1.Let G be a topological group acting on a topological space X.Given x 0 ∈ X, a local cross-section for the action of G on X at x 0 is a continuous map s : U → G defined on an open neighborhood U of x 0 satisfying: We say that the action of G on X admits local cross-sections if it does so at each point.
Up to replacing s(x) by s(x) • s(x 0 ) −1 in the above definition, we can assume that s(x 0 ) is the identity element.
Remark 2.2.It is well-known that, if X admits local cross-sections, then any G-equivariant map from a G-space to X is locally trivial, see e.g.Palais (1960, Theorem A).
The main result of this section adapts Mather's stability of smooth mappings (Mather, 1969) to the case of Morse functions with equal critical points by combining results of Cerf: The first result of Cerf we use is essentially a version of Proposition 2.3 restricted to the space Morse f (M ; Crit(f ), ∂M ) of Morse functions with the same critical points and critical values as f and the same value and derivatives of any order as f on ∂M .Proposition 2.4 (Cerf (1970), Appendix, §1, Proposition 1).Let f : M → R be a Morse function.Let G be the subspace of diffeomorphisms fixing the critical points of f and ∂M , and which have the same value and derivatives of every order as the identity on ∂M .Then the action of G on Morse f (M ; Crit(f ), ∂M ) admits local cross-sections.
To enhance this result to Morse functions with critical points and derivatives at the boundary allowed to vary, we need the reformulation of a result of Cerf (Cerf, 1961, Theorem 5) for the C ∞ case given in Hong et al. (2012).Proposition 2.5.Let N ⊆ M be a compact submanifold.Suppose that: (a) either N has no boundary and does not intersect the boundary ∂M ; (b) or N is a closed collar neighborhood of a boundary component ∂M j .Let L be a compact neighborhood of N .Let Emb L (N, M ) be the space of embeddings j : N ֒→ M such that j(N ) ⊆ L, j −1 (∂M ) = N ∩ ∂M , and that restricts to the identity on N ∩ ∂M .Denote by D L (M ) the space of diffeomorphisms inducing the identity on (M \ L).Then the action of D L (M ) on Emb L (N, M ) admits local cross-sections.
Proof.This is a direct consequence of Theorem 3.1 in Hong et al. (2012).Note that, strictly speaking, embeddings j as defined in (Hong et al., 2012, Definition 2.5) are required to admit an extension to a diffeomorphism of M .However this assumption is unnecessary for their Theorem 3.1 and so we omit it.
Proof of Proposition 2.3.Since g ∈ Morse f (M ) implies that Morse g (M ) = Morse f (M ), it is enough to construct a local cross-section at f .By Proposition 2.4, there exists a local cross-section g ∈ U f → φ g ∈ D Id (M ) defined on a neighborhood U f of f in Morse f (M ; Crit(f ), ∂M ), the space of functions with the same critical points p 1 , • • • , p n and critical values as f , and with the same value and derivatives ∂ k f as f on the boundary ∂M .Therefore For the general case where critical points and derivatives on the boundary are allowed to vary we simply find a diffeomorphism sending them back to p 1 , • • • , p n and ∂ k f and apply the above result.
Namely, from item (a) of Proposition 2.5, we can find disjoint compact neighborhoods In particular in this case g • ψ Crit(g) has the same critical points p 1 , • • • , p n as f , so it remains to deal with the boundary ∂M .
Let ∂M j be a boundary component.By flowing along the normalized gradient of f (or its inverse) from the boundary ∂M j we get a compact collar ) ⊆ [a j , a j + α].Therefore, after potentially shrinking V, we can continuously associate to g ∈ V the embedding ι Hence by using item (b) of Proposition 2.5, up to shrinking V, we can extend ι g to a diffeomorphism χ g that induces the identity outside V j .By repeating this process for each boundary component ∂M j , we can continuously associate to g ∈ V a diffeomorphism χ g such that g and f • χ g agrees on a closed collar neighborhood of the boundary.
By reducing the neighorhoods U i and V j to avoid overlaps, we have that for any g in U ∩V the Morse function g has the same critical points as f and agrees with f on a neighborhood of the boundary ∂M , in particular it belongs to Morse f (M ; Crit(f ), ∂M ).Hence by Eq. ( 1), up to shrinking U ∩ V, we have hence the local cross-section: Corollary 2.6.Let (f t ) t∈[0,1] ⊆ Morse(M ) be a path of Morse functions with the same critical values.Then there exists φ ∈

COVERING THE FIBER WITH DIFFEOMORPHISMS ISOTOPIC TO THE IDENTITY
Given f ∈ Morse(M ), we get a nested sequence of sub level-sets f −1 ((−∞, x]).In turn, by applying homology in degree 0 ≤ k ≤ d with coefficients in an arbitrary field, we get the persistent homology module of f : the sequence of vector spaces H k (f −1 ((−∞, x])) with linear maps between them induced by inclusions, in other words a functor from the poset (R, ≤) to finite dimensional vector spaces.The barcode of f in degree k is the isomorphism class of this functor up to natural isomorphism.From Crawley-Boevey ( 2015 and it is the zero vector space everywhere outside of [b, d).Therefore the barcode of f , denoted by PH k (f ), can be equivalently described as the multi-set D of pairs (b, d) indexing this decomposition, and will be described in this way in the rest of this document.By abuse of terminology we refer to pairs (b, d) as intervals or bars of the barcode We refer to Edelsbrunner and Harer (2008); Zomorodian and Carlsson (2005) for extensive treatments of the theory of Persistence.
In this work the persistence map is defined on Morse functions and returns the d + 1 barcodes of interest: We assume that Bar is equipped with its natural bottleneck metric which turns PH into a continuous map by the Stability Theorem (Cohen-Steiner et al., 2007).Proof.Let (φ t ) 0≤t≤1 be a path in D Id (M ).Each φ t restricts to a homeomorphism between the sub level-sets of f •φ t and f , hence it induces an isomorphism between the associated persistent homology modules.In turn PH(f Conversely let g ∈ PH −1 f (D) and let (f t ) 0≤t≤1 be a path in the fiber PH −1 f (D) joining f to g, thus PH(f t ) = D for each t.As is well-known, when M has no boundary there is a one-to-one correspondence between the set D of (bounded) interval endpoints in the barcode and the set C of critical values (counted with multiplicity) for Morse functions because the associated persistent homology module and Morse-Smale complex are isomorphic (Barannikov, 1994), see also (Leygonie et al., 2021, Proposition 2.14) for a self-contained proof.
When M has a boundary the correspondence adapts by adding in C the value a j with multiplicity i β i (∂M j ) for each boundary component ∂M j that is a local minimum.Note that a Morse function is constant on ∂M j and has no critical points there, so either it has ∂M j as a local minimum or as a local maximum, and this choice is fixed inside a path connected component of Morse(M ).
Therefore each f t has the same critical values as f , because the barcode PH(f t ) = D is constant.By corollary 2.6 there exists an Id-isotopic diffeomorphism φ such that g

TOPOLOGICAL PROPERTIES OF THE FIBER
We derive direct consequences of Theorem 3.1 combined with the extensive study of O Id (f ) by Maksymenko (2006).Strictly speaking, it is O f (f ), the path component of O(f ) containing f , whose properties are studied in Maksymenko (2006).However, there is an obvious inclusion O Id (f ) ⊆ O f (f ), and the reverse inclusion holds as well by corollary 2.6.Therefore O f (f ) = O Id (f ).
Denote by S Id (f ) the subspace of Id-isotopic diffeomorphisms φ preserving a Morse function f , i.e. f • φ = f . 1 Remark 4.2.The principal bundle S Id (f ) → D Id (M ) → PH −1 f (D) has computationally useful and direct implications.First, it is a locally trivial fibration hence it induces a homotopy long exact sequence: . Second, we have the homeomorphism: We apply this result to compute the path components of the fiber PH −1 (D) when M is a circle: Let n be the number of minima of f , which is then also the number of maxima of f because χ(S 1 ) = 0. Without loss of generality we assume that the associated 2n critical points of f are evenly spaced on S 1 .The space D Id (S 1 ) of Id-isotopic diffeomorphisms of the circle deformation retracts to S 1 , i.e. the rotations of the circle.The subgroup S Id (f ) of Id-isotopic diffeomorphisms φ preserving f , that is f • φ = f , is then (isomorphic to) the subgroup of rotations consisting of the 2n-th roots of unity that preserve the sequence of extremal values of f .The result follows since the quotient of S 1 by a finite subgroup is again S 1 .
When M = [0, 1] recall that Morse functions have prescribed values a 0 and a 1 on the boundary points 0 and 1.Note that we could easily derive a similar statement for Morse functions on [0, 1] without boundary conditions.In Section 5 we prove the analogues of Propositions 4.3 and 4.4 for continuous functions.The analogues for lower-star filtrations on the subdivided interval and circle have been proved by Cyranka et al. (2020) and Mischaikow and Weibel (2021) respectively.
For this reason we focus our analysis to the interesting case where M is connected.
For the rest of the section we fix a compact connected surface M and a function f with barcode D, whose number of critical points of index 1 is denoted by c 1 .We make use of the analysis of the orbit O Id (f ) by Maksymenko (2006).Remark 4.7.From Maksymenko (2006, (2), Theorem 1.5) we can also derive a short exact sequence where G is a finite group and the integer k f ≥ 0 depends on the component PH −1 f (D) in the fiber, on the number c 1 of saddles and the surface M .
Proposition 4.8.Assume that c 1 = 0. Then the homotopy type of the fiber For instance the case where f : S 2 → R has no saddle (c 1 = 0) can be interpreted as follows: The fiber sequence S Id (f ) → D Id (S 2 ) → PH −1 f (D) of Proposition 4.1 can be identified up to homotopy with the standard fiber sequence S 1 → SO(3) → S 2 .This is because D Id (S 2 ) deformation retracts to SO(3) by the 2-dimensional Smale conjecture (see Smale (1959)), and if without loss of generality we assume that f is the standard height function, then S Id (f ) consists of those rotations fixing the poles, so S Id (f ) is fixed by the retraction and S Id (f ) ∼ S 1 .Proposition 4.9.Assume that D has pairwise distinct bounded interval endpoints, and that c 1 > 0. Then we have the following homotopy types for the fiber When M is obtained from the surfaces in the above tables by removing finitely many 2-disks, then M) .For the remaining non-orientable surfaces, we have When ∂M = j ∂M j = ∅, we can partition the boundary components ∂M j into the sets ∂M min (resp.∂M max ) of components ∂M j that are local minimum (resp.maximum) of one (hence any) function f in the component of PH −1 (D) at stake.Since M is a surface each ∂M j is a circle, therefore if ∂M j ⊆ ∂M min , then it corresponds in the barcode D to the births of one interval in degree 0 and one interval in degree 1. Otherwise ∂M j ⊆ ∂M max induces no topological change when entering the sub level-sets of f .Consequently the correspondence between critical points and interval endpoints adapts and yields c Remark 4.11.For manifolds M of dimension 3 for which the Smale conjecture D(M ) ∼ = Isom(M ) holds, e.g. the 3sphere, lens spaces, prism and quaternionic manifolds (see Hong et al. (2012)), the homotopy type of D Id (M ) is quite well-understood.For instance we have D Id (S 3 ) ∼ = SO(4).However, to deduce the homotopy groups of PH −1 f (D), we lack the understanding of less-studied topological properties of S Id (f ).

FIBER OF PERSISTENT HOMOLOGY FOR CONTINUOUS MAPS ON THE CIRCLE AND ON THE INTERVAL
In this section the domain of the persistence map consists of continuous maps on the circle: Note that in the codomain we record the two barcodes with non-trivial homology, those in degree 0 and 1.In fact the second barcode contains a unique unbounded interval starting at the maximum of the function on the circle.We fix a barcode D with finitely many intervals.When f = cst is constant it forms the fiber by itself over the trivial barcode D = PH(f ) with only two infinite bars (cst, +∞), one in each degree 0 and 1.Other barcodes such that PH −1 (D) = ∅ have one infinite interval (b 0 , +∞) in degree 0, one infinite interval (b 1 , +∞) with b 0 < b 1 in degree 1, finitely many bounded intervals in degree 0 with endpoints in [b 0 , b 1 ], and no other intervals.In the rest of this section we assume that D is non-trivial and denote by (n − 1), for some n ≥ 1, its number of bounded intervals in degree 0.
Let Aut ≤ (S 1 ) be the space of orientation-preserving homeomorphisms of the circle, and End ≤ (S 1 ) = Aut ≤ (S 1 ) be its closure in C 0 (S 1 , R) in the compact-open topology.Given f ∈ C 0 (S 1 , R) we have the pre-composition map φ ∈ End ≤ (S 1 ) → f • φ ∈ C 0 (S 1 , R); we denote by S Id (f ) the stabiliser of f and by O Id (f ) its orbit.
Proposition 5.1.The fiber PH −1 (D) has finitely many path connected components.In each such component Ω(D) there exists some f Ω(D) : S 1 → R such that: ), and then Ω(D) is homeomorphic to the quotient End ≤ (S 1 )/S Id (f Ω(D) ), and in particular is homotopy equivalent to S 1 .
Unlike the smooth case the component Ω(D) ⊆ PH −1 (D) in the fiber equals the orbit of a function only for a careful choice of function f Ω(D) : the requirement will be that f Ω(D) is injective between its consecutive extrema.Nevertheless the fact that the pre-composition map induces a homeomorphism from End ≤ (S 1 )/S Id (f Ω(D) ) to the orbit O Id (f Ω(D) ) is reminiscent of the smooth case, and in fact with slightly more work it can be shown that it defines a S Id (f Ω(D) )-principal bundle.We state without proof the analogous and simpler result for the unit interval [0, 1], which works with or without fixed values on the boundary points 0 and 1. Proposition 5.2.For any finite barcode D the fiber PH −1 (D) ⊆ C 0 ([0, 1], R) has finitely many path connected components, each of which is contractible.
Using a fixed orientation on S 1 and going around starting from the north pole we can order the n minima and n maxima of a non-constant f ∈ C 0 (S 1 , R) into a sequence Val(f ) which we view as an element in R 2n : Associated to this sequence we have the sequence of critical sets of f : . Then f has 2n extrema, i.e.Val(f ) ∈ R 2n .In addition, let Γ n be the group of cyclic permutations on n elements, which acts on R 2n by cyclically permuting the n pairs of entries.Then the connected component Ω(D) in the fiber containing f is made of functions g whose sequence of extrema is the same as that of f up to a different ordering, that is: We omit the proof of this elementary statement.So if Ω(D) is a component in the fiber, we can pick the following simple function f Ω(D) in Ω(D), whose critical sets and extrema are denoted by c i , d i , m i , M i for simplicity: the critical sets c i and d i are singletons arranged on the regular 2n-gon in S 1 and on each circular arc [c i , d i ], f Ω(D) restricts to the linear homeomorphism to [m i , M i ].
Finally we show that the inverse is continuous.Let f ∈ Ω(D) and φ f,π as in (3).Up to pre-composing f by a suitable homeomorphism the north pole does not belong to any extremal set c i (f ), d i (f ).Consequently, for g in a small neighborhood U ⊆ Ω(D) around f , we also have Val(g) = π.Val(f Ω(D) ), hence we can define φ g,π ∈ End ≤ (S 1 ) like in Eq. ( 3) and then f Ω(D) • φ g,π = g.Hence the map g ∈ U −→ φ g,π ∈ End ≤ (S 1 ) is a local section, whose continuity is a consequence of the fact that on each circular arc [c i , d i ] the linear restriction (f Ω(D) ) |[ci,di] and its inverse are Lipschitz, and of the fact that the maximal distance from points in the critical sets c i (g), d i (g) to the critical sets c i (f ), d i (f ) of f can be continuously tracked in a sufficiently small neighborhood U of f , see Fig. 1.The technical details are omitted.
A piece of a continuous function f : S 1 → R and a small neighborhood U indicated by dashed curves.Any function in PH −1 (D) between the dashed curves must have a critical value in each U i and V i , provided the band between the dashed curves is thin enough to separate critical values.If g is such a function then these must be the only critical values.Then the critical value of g in U i must be m i , in V i must be M i , and so on.
Proof of Proposition 5.1.From Proposition 5.4 the pre-composition map φ → f Ω(D) • φ induces a homeomorphism from End ≤ (S 1 )/S Id (f Ω(D) ) to the path connected component Ω(D).Besides it is well-known that End ≤ (S 1 ) deformation retracts to the group SO(2) ∼ = S 1 of orientation preserving rotations. 2Recall that f Ω(D) is a piece-wise linear interpolation between extremal values arranged on a regular 2n-gon, therefore its stabiliser S Id (f Ω(D) ) is a finite subgroup of SO( 2) which is preserved under the deformation retraction.Hence Ω(D) is homotopy equivalent to the quotient of SO(2) ∼ = S 1 by a finite subgroup, so it is in fact homotopy equivalent to S 1 .

Proposition 2 . 3 .
Let f ∈ Morse(M ) and Morse f (M ) be the subspace of Morse functions with the same critical values as f .Then the action of D Id (M ) on Morse f (M ) admits local cross-sections.
) any such functor uniquely decomposes as a direct sum of functors (b,d)∈D I [b,d) , with [b, d) ⊆ R an interval closed on the left and open on the right (hence possibly d = +∞): each I [b,d) consists of 1-dimensional vector spaces linked with identity maps on [b, d), Given a barcode D and a Morse function f ∈ Morse(M ) such that PH(f ) = D, we denote by PH −1 f (D) the path connected component of the fiber PH −1 (D) ⊆ Morse(M ) containing f .Theorem 3.1.Let D be a barcode and f ∈ PH −1 (D).Then PH −1 f (D) = O Id (f ).

Proposition 4. 1 .
Assume that M is connected.Let D be a barcode and f ∈ PH −1 (D).Then the action of D Id (M ) on PH −1 f (D) defines a locally trivial principal S Id (f )-fibration.Proof.From Maksymenko (2006, Theorem 2.1, (2)) the action of diffeomorphisms D(M ) on O(f ) defines a locally trivial principal fibration with fiber the diffeomorphisms φ satisfying f • φ = f .Restricting to the action of D Id (M ) on O Id (f ) defines a locally trivial principal S Id (f )-fibration, and PH −1 f (D) equals O Id (f ) by Theorem 3.1.

Remark 4. 5 .
When M = M 1 ⊔ M 2 has more than one connected component, the path component PH −1 f (D) in the fiber over D = PH(f ) can be retrieved as the product of the path components of the fibers over D 1 := PH(f |M1 ) and D 2 := PH(f |M2 ) containing the restrictions f |M1 and f |M2 respectively:
by Theorem 3.1, and the homotopy type of O Id (f ) is computed byMaksymenko (2006,  Theorem 1.9).
by Theorem 3.1.Since D has distinct bounded interval endpoints, f has distinct critical points, and then the homotopy type of O Id (f ) is computed by Maksymenko (2006, (2)&(3), Theorem 1.5).Remark 4.10.When M has no boundary, ∂M = ∅, the number c 1 of saddles of Morse functions f in the fiber PH −1 (D) can be directly inferred from the barcode D. Namely, if we denote by k D the number of intervals in D, then the quantity k D −β 0 −β 2 counts (i) all the intervals (b, d) of D in degree 1, which correspond by their birth value b to saddle points of f whose attaching handle increases the 1-dimensional homology of the sub level-set f −1 ((−∞, b]), and (ii) all the bounded intervals (b, d) of D in degree 0, which correspond by their death value d < ∞ to saddle points of f whose attaching handle decreases the 0-dimensional homology of the sub level Work supported by the Mathematical Institute of Oxford & the EPSRC grant EP/R018472/1.Both authors are members of the Centre for Topological Data Analysis.