Sards theorem topological analysis approach

Transversality and intersection theory / Sard’s theorem, topological analysis approach
Introduction
The concept of transversality deals with the intersection of two objects; in several ways, one may consider it the reverse of tangency. For transversality to occur between two sub- manifolds, their tangential spaces at every intersection point need to extend across the ambient manifold’s tangent space. Transversality, invariably, particularly fails in case of tangency between two sub- manifolds. However, a more notable point is, tangency lacks stability: all situations involving tangency between two objects may be effortlessly and somewhat disturbed into non- tangent situations, which isn’t true when it comes to transversality. Part of the reason why transversality is so sound a tool is its stability.
Rene Thom, a French mathematician, introduced the idea of transversality during the 50s. In his doctoral thesis performed in the year 1954, he included the proof and statement of his Transversality Theorem (Greenblatt, 2015), which proves transversality’s generic nature (i.e., all non- transverse intersections may be deformed (through small arbitrary deformations) into transverse intersections). This property is sounder as compared to stability.
Theory of Transversality
In the domain of mathematics, transversality represents a concept describing intersection between spaces; it may be perceived to be the reverse of tangency, contributing to general position. The theory formalizes the concept of a broad intersection within differential topology and is described through taking into consideration intersecting spaces’ linearizations at intersection points. Surface arcs form a non- trivial and most basic example of the phenomenon. Intersection points between arcs are transverse iff they aren’t tangencies (in other words, in the event of distinct tangent lines within the surface’s tangent plane). Transverse curves fail to intersect in 3D spaces (Thom, 1954). Curves that are transverse to a surface will intersect one another in points, whilst surfaces that are transverse to one another will intersect in the form of curves. Curves tangent to surfaces at any given point (e.g., curves that lie on a given surface) don’t transversally intersect surfaces.
When the value y is regular, f?1 (y) is manifold. The above statement is ‘generalizable’ to complete co- domain subsets Z, so long as the transversality condition is met with.
Definition: All smooth functions, f : X ? Y, are transverse to sub- manifolds Z ? Y at x if:
Im (dfx) + Tf(x) (Z) = Tf(x) (Y )
In other words, all components of Tf(x) (Y) may be considered the sum of components in Tf(x) (Z) and Im (dfx).
Example: From X = R, f (t) = (0, t), Y = R2, and Z = < 1, 0 >, a transverse mapping is obtained. The transversality is because: Im (df) = < 0, 1 >, and Z + Im (df) = span (e1, e2) = R2.
A special case would be: X, Z ? Y being sub- manifolds with f: X ? Y taken as inclusion. According to the transversality condition:
Tp (X) + Tp (Z) = Tp (Y )
In other words, in case of intersection of X and Z at any point, their tangential spaces at the point need to span those of Y. Two manifolds are said to transversally intersect Z ? X in the event the above condition is fulfilled.
Theorem: If f: X ? Y is transverse to Z ? Y (sub- manifold), f?1 (Z) is also a sub- manifold.
Stability
A key facet of the analysis of maps’ properties is the stability of the properties in instances of slight deformations.
Definition: Two maps f0, f1: X ? Y may be considered smoothly homotopic in the event some F : X × [0, 1] ? Y such that F (x, 0) = f0 (x) and F (x, 1) = f1 (x).
Definition: P (a property of maps) may be deemed to be stable in the event it remains unchanged when subject to slight deformation. In specific, if f0 : X ? Y is able to fulfill stable property (i.e., P) with F being some homotopy with F(x, 0) = f0, there exists ? > 0 such that F (x, ?) = f? fulfills P for every ? < ?.
Theorem: Stable map properties on manifolds are listed below:
(a) immersion
(b) local diffeomorphism
(c) embedding
(d) submersion
(e) diffeomorphism
(f) transverse to some sub- manifold Z
Intersection Theory
Orientation of Manifolds
Defining manifold orientations requires firstly defining it on a vector space.
Vector Space (V)
Consider {vi} and {ui} to be the ordered bases of any given vector space (i.e., V). Change in basis matrix (i.e., P) maps two bases, inducing equivalence relations on the sets of the vector space’s ordered bases ?, ?’ equivalent when det (P) > 0.
Definition: Vector space orientations may be defined as the map B ? {± 1}, (B being a set of ordered bases, with equivalent bases sharing an identical sign). A vector space isomorphism A is said to be orientation preserving in case ? ? ? 0 ? A? ? A?’ (if not, it is termed orientation reversing).
Manifolds
Manifold orientations are grounded in vector space orientations:
Definition: Orientations of bounded smooth manifolds represent a smooth orientation choice on every Tx (X). In this context, the term ‘smooth’ implies that around all x ? X, a parameterization, i.e., h : U ? X must exist such that dh : Rk ? Th(u) (X) becomes orientation preserving (Rk has normal orientation).
Remark: Every manifold doesn’t admit orientations (e.g., he M¨obius Strip). In case an orientation exists for manifold X, it will inevitably have another orientation, ?X – the converse basis choice at all points.
X and Y (two sub- manifolds) within the ambient space M may be said to transversally intersect each other if, for every,

Here, the addition is within ;  represents’s tangent map. Two sub- manifolds will automatically be transversal in the event they fail to intersect. For instance, a couple of curves in  will be transversal only in case they entirely fail to intersect. Transversal meeting of  and  means that  denotes a ‘smooth’ sub- manifold of (the anticipated dimension)
In a way, two sub- manifolds should transversally intersect; furthermore, according to Sard's theorem, all intersections may be perturbedly transversal. Homological intersection only seems sensible since intersections may be rendered transversal (Sard, 1942).
Transversality suffices for stability of intersections following perturbation. For instance, the lines and  as well as (perturbed lines) transversally intersect, at a single point only. But,  and don’t transversally intersect. They do so in a single point, whereas  intersects in a couple of points or no points at all (this depends on whether or not  is negative).
If , transversal intersections become isolated points. In case of vector space orientations of the spaces, the transversality condition allows assigning of signs to the intersections. When  form oriented bases for , with   being oriented bases for , the intersection becomes  when  is oriented in , or else it is .
In broader terms, the smooth maps and  will be transversal in the event that, whenever , .
It is said that two sub- manifolds of any finite- dimensioned smooth manifold transversally intersect if, at all intersection points, their individual tangent spaces at the point combined generate ambient manifold tangent spaces at the point (Thom, 1954). Non- intersecting manifolds are vacantly transverse. In case of complementary dimensional manifolds (that is, dimensions adding up to ambient space dimensions), the condition implies tangent spaces to ambient manifolds are directly the sum of smaller tangential spaces. In case of transverse intersections, the intersection is a sub- manifold with co- dimension equivalent to the sum of the two manifolds’ co- dimensions. Without transversality, the intersection might not be a sub- manifold, and have a kind of singular point.
Sard’s Theorem
The pre- images of regular values, y, of smooth mappings f: X ? Y will be smooth manifolds, raising the natural question of:
Q: What quantity of Y constitutes regular values?
A: Sard’s theorem solves this issue. For stating this, the notion of measure on manifolds must be defined.
Definition: Any subset A ? R? is said to have a measure of zero if it is ‘coverable’ by a finite number of arbitrarily small rectangles.
Definition: A subset A of manifold M (i.e., A ? M) is said to have measure zero iff:
1. For all parameterizations ? : U ? M, the pre- images ??1 (A) have a measure of zero as subset of R?.
2. A covering for M exists by charts (??, U?) such that ??1 ? (A) is characterized by measure of zero as subset of R? for every ?.
Theorem (Sard’s): If f: X ? Y represents a smooth manifold map, nearly all points in Y are regular. In other words, if C forms a set of its critical points, f (C) has a measure of zero.
Importance and application of Sard’s theorem
Sard's theorem has chiefly been applied for escaping from a function’s critical values, through slight perturbation. Additionally, it helps extend definitions only valid at typical values, to the critical values. Homotopy lemma represents the most basic example within intersection theory. If f: M ? N is a smooth map between a couple of manifolds of a given dimension, with NN being connected and M being compact. The intersection number ff at point y ? N denotes number of elements within the set f? 1(y). When y is any regular value, the figure remains identical for every regular value in its local neighborhood. The intersection number may be defined when y represents any critical value through setting it at the same value as intersection number following perturbation of y to an almost regular value, existing by Sard's theorem (Sard, 1942). According to the Homotopy Lemma, two homotopic functions share a common intersection number modulo 2. Once again, the evidence banks on Sard's theorem for escaping critical values.
According to Sard's theorem, a smooth function’s critical value set is tiny (i.e., a set of measure zero). Simply put, it states that functions can’t have an overly large number of local minimum and maximum values. Consider the function  f: R ? R,  f (x) =sin (x) with range [?1, 1]. The range has two special values: {?1, 1}. If f adopts one value, the derivative will be zero. These are critical values of the function. The critical values form a set of measure of zero in R. Consider the more complex function, f (x) = e?x sin (x) with critical values being {± e ±?/2, ± e ±3?/2, ± e ±5?/2?} (i.e., a countably infinite quantity though nevertheless discrete and having a measure of zero in R).
Topological analysis approach
TDA or Topological Data Analysis represents a new domain which stemmed from multiple works in the areas of computational geometry and applied topology (algebraic) in the 2000s. While data analysis’s geometric approaches date back rather far, TDA actually took the shape of a discipline with Carlsson and Zomorodian’s (2005) pioneering works in the area of persistent homology. Further, Carlsson’s (2009) breakthrough work popularized the field. TDA is largely inspired by the notion that geometry and topology offer an effective approach for inferring sound qualitative, and even quantitative, details on information structure (Chazal & Michel, 2017).
TDA endeavors to provide sound statistical, algorithmic, and mathematical techniques for inferring, analyzing and applying the multifaceted geometric and topological structures that underlie information typically represented in the form of point clouds within Euclidean or broader metric spaces. In the past few years, tremendous efforts have been expended towards offering sound, effective algorithms and data structures for TDA (made user- friendly by standard libraries like the Gudhi library (Python and C++) (Maria et al., 2014) and the R software interface (Fasy et al., 2014). While it continues to swiftly evolve, TDA currently offers a collection of advanced, effective tools which may be utilized complementary to or together with other instruments in the field of data sciences.
The TDA pipeline
Of late, TDA has advanced in several fields of application and directions. Various techniques that are prompted by geometric and topological approaches may be found. To offer a comprehensive summary of all approaches lies beyond this paper’s scope. But a majority of them are reliant on the standard, fundamental pipeline described below:
1. It is supposed that the input constitutes a fixed collection of points that come with the idea of similarity or gap between them. The gap may be prompted by the system of measurement in ambient space (for instance, Euclid’s metric if information is imbedded in Rd) or emerge as the inherent metric described using a paired distance matrix. Its definition on the data is normally application- driven or furnished as input (Chazal & Michel, 2017). But a key point to note is that the metric selected can be crucial when it comes to revealing interesting data- related geometric and topological elements.
2. An uninterrupted shape is constructed atop the data for highlighting the fundamental geometry or topology. This is usually a simplicial complex or its nested family (termed filtration) which reflects the data structure at multiple scales. A simplicial complex may be deemed to be a higher dimensional generalization of adjacent graphs built characteristically atop data in several regular learning or data analysis algorithms. Here, the challenging thing is delineating structures demonstrated to reveal pertinent information concerning information structure and which may be successfully built and practically manipulated (Chazal & Michel, 2017).
3. Geometric or topological data is derived from structures constructed atop the information, either leading to a complete rebuilding of the information’s fundamental shape (normally a triangulation), wherefrom geometric or topological features may conveniently be derived or, in rough estimates or outlines from which obtaining appropriate data calls for certain distinct techniques like continual homology (Chazal & Michel, 2017). Beyond identifying geometric or topological data of interest and its conceptualization and analysis, the challenging task here is demonstrating its applicability, especially its stability in connection with perturbations or the existence of noise within input information. To this end, a key question is grasping inferred elements’ statistical conduct.
4. The geometric and topological data obtained presents novel families of information descriptors and elements which may be utilized for better understanding information, especially via conceptualization, or may be used together with other types of elements for additional examination and mechanized learning activities. Proving the relative complementarity of data (to other elements) and the additional value, offered by TDA instruments, is a key question here.
TDA and statistics
Until recent times, TDA’s topological inference and conceptual facets were largely reliant upon deterministic strategies which failed to consider the randomness of the information and the essential inconsistency of the inferred topological measure. As a result, a majority of corresponding techniques continue to be exploratory, and fail at effectively distinguishing between “topological noise” and information (Chazal & Michel, 2017).
Adopting a statistical method to deal with TDA implies perceiving information to result from an unidentified distribution; additionally, it implies topological facets deduced using TDA techniques are considered to be estimators of the topological measures that describe a fundamental entity. As part of this approach, indefinite objects are generally consistent with information distribution support (or, at the very least, near such support). But it doesn’t invariably have a tangible existence (e.g., galaxies within our universe are arranged along filaments which aren’t tangible in nature).
The chief aims of statistical means of approaching topological information examination may be summed up as the list of issues delineated below:
Topic 1: Establishing constancy and examining TDA techniques’ convergence rates.
Topic 2: Offering confidence areas for topological elements and exploring the importance of projected topological measures.
Topic 3: Choosing suitable scales for the consideration of topological phenomena, as functions of observed information.
Topic 4: Handling outliers and offering sound TDA techniques (Chazal & Michel, 2017).
Lessons learnt as an undergrad senior math student
Transverality has been noted to be a rather fine condition: two objects once transverse will typically continue to be so when subject to minor perturbations; better still, non- transverse objects may, at any given time, be perturbed without difficulty and rendered transverse. The concept of transversality proves rather valuable when it comes to extending the Pre- image Theorem and stating when a manifold’s pre- image (rather than only one single point) in smooth maps is a manifold as well. Moreover, transversality’s genericity as well as stability renders it a strong criterion.
Transversality’s genericity and stability, as mentioned above, render it a rather robust condition, resulting in numerous applications, in multiple scientific branches, that may, at first, not appear to be associated with differentiable manifolds. For data sets that may be represented in the form of manifolds transverse to certain important conditions, no trivial data set perturbations will impact its relationship with the important condition.
Topologically ND (non- degenerate) Functions: A noteworthy fact here pertains to the mathematical probabilities that potentially focus on the singular homology theory’s development without employing triangulation or CW type block subdivision. This is advantageous as well as practical, particularly if one bears in mind the fact that the areas of examination to which we naturally apply the critical point theory generally fail to be locally compact; further, non- isotopic retractions and deformations are expected instruments, especially within variational theory.
From Sard’s definition of topologically non- degenerate functions and their utilization, coupled with the techniques Morse applied in his lectures (Morse, 1947), it is rather clear that: If M (any small topological manifold) has a topologically non- degenerate function f, one can carry through the determination of Mn’s singular homology groups, up to an isomorphism, which will basically lead to identical homological relations as within the differentiable instance.
Compact triangulable topological manifolds exist that do not admit any differentiable structure. If, additionally, a compact combinatorial manifold becomes a topologically non- degenerate function’s domain, a key question will be: Do topological manifolds I‘n exist that aren’t triangulable and don’t admit differentiable structures, though being domains of any topologically non- degenerate function, f? Homology groups of these manifolds can then be determined using an explicitness earlier believed to be impossible.

















References
Carlsson, G. (2009). Topology and data. AMS Bulletin, 46(2):255–308.
Chazal, F., & Michel, B. (2017). An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists. arXiv preprint arXiv:1710.04019.
Fasy, B. T., Kim, J., Lecci, F., & Maria, C. (2014). Introduction to the R package TDA. arXiv preprint arXiv:1411.1830.
Greenblatt, C. (2015). An introduction to Transversality. Accessed online at http://schapos. people. uic. edu/MATH549_Fall2015_files, 1.
Maria, C., Boissonnat, J. D., Glisse, M., &Yvinec, M. (2014, August). The gudhi library: Simplicial complexes and persistent homology. In International Congress on Mathematical Software (pp. 167-174). Springer, Berlin, Heidelberg.
Morse, M. (1947). Introduction to analysis in the Large. 1947 Lectures.
Sard, A. (1942). The measure of the critical values of differentiable maps. Bulletin of the American Mathematical Society, 48(12), 883-890.
Thom, R. (1954). Quelquespropriétésglobales des variétésdifférentiables. CommentariiMathematiciHelvetici, 28(1), 17-86.
Zomorodian, A., & Carlsson, G. (2005). Computing persistent homology. Discrete & Computational Geometry, 33(2), 249-274.