Tag Archives: Topology

Echelon Form Lemma (Column Echelon vs Smith Normal Form)

The pivots in column-echelon form are the same as the diagonal elements in (Smith) normal form. Moreover, the degree of the basis elements on pivot rows is the same in both forms. Proof: Due to the initial sort, the degree … Continue reading

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Persistent Homology Algorithm

Algorithm for Fields In this section we describe an algorithm for computing persistent homology over a field. We use the small filtration as an example and compute over , although the algorithm works for any field. A filtered simplicial complex … Continue reading

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De Rham Cohomology

De Rham Cohomology is a very cool sounding term in advanced math. This blog post is a short introduction on how it is defined. Definition: A differential form on a manifold is said to be closed if , and exact … Continue reading

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Singular Homology

A singular -simplex in a space is a map . Let be the free abelian group with basis the set of singular -simplices in . Elements of , called singular -chains, are finite formal sums for and . A boundary … Continue reading

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Mapping Cone Theorem

Mapping cone Let be a map in . We construct the mapping cone , where is identified with for all . Proposition: For any map we have if and only if has an extension to . Proof: By an earlier … Continue reading

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Tangent space (Derivation definition)

Let be a smooth manifold, and let . A linear map is called a derivation at if it satisfies The tangent space to at , denoted by , is defined as the set of all derivations of at .

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Homology Group of some Common Spaces

Homology of Circle Homology of Torus Homology of Real Projective Plane Homology of Klein Bottle Also see How to calculate Homology Groups (Klein Bottle).

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Summary of Persistent Homology

We summarize the work so far and relate it to previous results. Our input is a filtered complex and we wish to find its th homology . In each dimension the homology of complex becomes a vector space over a … Continue reading

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Structure Theorem for finitely generated (graded) modules over a PID

If is a PID, then every finitely generated module over is isomorphic to a direct sum of cyclic -modules. That is, there is a unique decreasing sequence of proper ideals such that where , and . Similarly, every graded module … Continue reading

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Persistence Interval

Next, we want to parametrize the isomorphism classes of the -modules by suitable objects. A -interval is an ordered pair with . We may associate a graded -module to a set of -intervals via a bijection . We define for … Continue reading

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Homogenous / Graded Ideal

Let be a graded ring. An ideal is homogenous (also called graded) if for every element , its homogenous components also belong to . An ideal in a graded ring is homogenous if and only if it is a graded … Continue reading

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Persistence module and Graded Module

We show that the persistent homology of a filtered simplicial complex is the standard homology of a particular graded module over a polynomial ring. First we review some definitions. A graded ring is a ring (a direct sum of abelian … Continue reading

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Persistence module and Finite type

A persistence module is a family of -modules , together with homomorphisms . For example, the homology of a persistence complex is a persistence module, where maps a homology class to the one that contains it. A persistence complex (resp.\ … Continue reading

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Homotopy for Maps vs Paths

Homotopy (of maps) A homotopy is a family of maps , , such that the associated map given by is continuous. Two maps are called homotopic, denoted , if there exists a homotopy connecting them. Homotopy of paths A homotopy … Continue reading

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Universal Property of Quotient Groups (Hungerford)

If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely. This can be used to prove the following proposition: A chain map between chain complexes and induces homomorphisms … Continue reading

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Some Homology Definitions

Chain Complex A sequence of homomorphisms of abelian groups with for each . th Homology Group is the free abelian group with basis the open -simplices of . -chains Elements of , called -chains, can be written as finite formal … Continue reading

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Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

Motivation Data is commonly represented as an unordered sequence of points in the Euclidean space . The global `shape’ of the data may provide important information about the underlying phenomena of the data. For data points in , determining the … Continue reading

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Natural Equivalence relating Suspension and Loop Space

Theorem: If , , , , Hausdorff and locally compact, then there is a natural equivalence defined by , where if is a map then is given by . We need the following two propositions in order to prove the … Continue reading

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Fundamental Group of S^n is trivial if n>=2

if We need the following lemma: If a space is the union of a collection of path-connected open sets each containing the basepoint and if each intersection is path-connected, then every loop in at is homotopic to a product of … Continue reading

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Functors, Homotopy Sets and Groups

Functors Definition: A functor from a category to a category is a function which – For each object , we have an object . – For each , we have a morphism Furthermore, is required to satisfy the two axioms: – For … Continue reading

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Algebraic Topology: Fundamental Group

Homotopy of paths A homotopy of paths in a space is a family , , such that (i) The endpoints and are independent of . (ii) The associated map defined by is continuous. When two paths and are connected in this way … Continue reading

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Topological Monster: Alexander horned sphere

Very interesting object indeed. Also see this previous video on How to Unlock Interlocked Fingers Topologically? The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is simply connected; i.e., every loop can … Continue reading

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Hatcher 2.1.6

Compute the simplicial homology groups of the -complex obtained from 2-simplices by identifying all three edges of to a single edge, and for identifying the edges and of to a single edge and the edge to the edge of . … Continue reading

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Wonderful Topology Notes for Beginners

Recently found a wonderful topology notes, suitable for beginners at: http://mathcircle.berkeley.edu/archivedocs/2010_2011/lectures/1011lecturespdf/bmc_topology_manifolds.pdf It starts by pondering the shape of the earth, then generalizes to other surfaces. It also has a nice section on Fundamental Polygons and cutting and gluing, which was what … Continue reading

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From Triangle to Mobius Strip

What familiar space is the quotient -complex of a 2-simplex obtained by identifying the edges and , preserving the ordering of vertices? This is a question from Hatcher. So if we “glue” two edges of the triangle together, preserving order, … Continue reading

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Covering map is an open map

We prove a lemma that the covering map is an open map. Let be open in . Let , then has an evenly covered open neighborhood , such that , where the are disjoint open sets in , and is … Continue reading

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Topology Puzzle

Assume you are a superman who is very elastic, after making linked rings with your index fingers and thumbs, could you move your hands apart without separating the joined fingertips? In other words, is it possible to go from (a) to (b) … Continue reading

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Topology book for General Audience

Previously I blogged about the book The Shape of Space, here is another book that is also about topology and suitable for a general audience (high school and above)! Intuitive Topology (Mathematical World, Vol 4)

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Effective Homotopy Method

The main idea of the effective homotopy method is the following: given some Kan simplicial sets , a topological constructor produces a new simplicial set . If solutions for the homotopical problems of the spaces are known, then one should … Continue reading

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Representing map of x

Proposition: (Representing map of ). Let be a simplicial set and let . There exists a unique simplicial map such that . Proof: Let . can be written as iterated compositions of faces and degeneracies of , i.e. where is … Continue reading

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Minimal Simplicial Sets

Let be a space. We can have a fibrant simplicial set, namely the singular simplicial set , where is the set of all continuous maps from the -simplex to . However seems too large as there are uncountably many elements … Continue reading

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Higher Homotopy Groups

We can generalise the idea to higher homotopy groups as follows. Let be a pointed fibrant simplicial set. The fundamental group is the quotient set of the spherical elements in subject to the relation generated by if there exists such … Continue reading

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Associativity and Path Inverse for Fundamental Groupoids

Continued from Path product and fundamental groupoids (Associativity). Let be a fibrant simplicial set and let , and be paths in such that and . Then Let be a point. Denote as the constant simplicial map for . (Path Inverse). Let … Continue reading

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Path product and fundamental groupoids

Let . A path is a simplicial map . Since , the paths are in one-to-one correspondence to the elements in via the function . The initial point of is , and the end point of is . Let and … Continue reading

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Homotopy Groups

Let be a pointed fibrant simplicial set. The homotopy group , as a set, is defined by , i.e.  the set of the pointed homotopy classes of all pointed simplicial maps from to . as sets. An element is said … Continue reading

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Geometrical Meaning of Matching Faces

Let be a simplicial set. The elements are said to be matching faces with respect to if for and . Geometrically, matching faces are faces that “match” along lower-dimensional faces. In other words, they are “adjacent”. In the 2-simplex, let … Continue reading

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Fibrant Simplicial Set

Let be a simplicial set. Then is fibrant if and only if every simplicial map has an extension for each . Assume that is fibrant. Let . The elements are matching faces with respect to . This is because for … Continue reading

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Motivation of Simplicial Sets

A simplicial set is a purely algebraic model representing topological spaces that can be built up from simplices and their incidence relations. This is similar to the method of CW complexes to modeling topological spaces, with the critical difference that … Continue reading

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Homotopy Theory on Simplicial Sets

Let be simplicial maps. We say that is homotopic to (denoted by ) if there exists a simplicial map such that and for all . If is a simplicial subset of and are simplicial maps such that , we say … Continue reading

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Wedge and Smash Products

Let and be pointed simplicial sets. The wedge of and , denoted by , is the simplicial set obtained by identifying the basepoint of with the basepoint of . Hence, we can define by the push-out diagram can be described … Continue reading

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Push-out

Let and be simplicial maps. We define to be the push-out in the diagram i.e. , where is the equivalence relation such that for . Then with faces and degeneracies induced from that in and forms a simplicial set with … Continue reading

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The Shape of Space

Just came across this book: The Shape of Space (Chapman & Hall/CRC Pure and Applied Mathematics). It is a very unique book, in the sense that it is aimed at high school students, but even a undergraduate or graduate student … Continue reading

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Behavior of Homotopy Groups with respect to Products

This blog post is on the behavior of homotopy groups with respect to products. Proposition 4.2 of Hatcher: For a product of an arbitrary collection of path-connected spaces there are isomorphisms for all . The proof given in Hatcher is … Continue reading

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The Fundamental Group

Source: Topology (2nd Economy Edition) If we pick a point of the space X to serve as a “base point” and consider only those paths that begin and end at , the set of these path-homotopy classes does form a … Continue reading

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Proof of Associativity of Operation * on Path-homotopy Classes

(Continued from https://mathtuition88.com/2015/06/25/the-groupoid-properties-of-operation-on-path-homotopy-classes-proof/) Earlier we have proved the properties (2) Right and left identities, (3) Inverse, leaving us with (1) Associativity to prove. For this proof, it will be convenient to describe the product f*g in the language of positive … Continue reading

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The Groupoid Properties of * on Path-homotopy Classes

This is one of the first instances where algebra starts to appear in Topology. We will continue our discussion of material found in Topology (2nd Economy Edition) by James R. Munkres. First, we need to define the binary operation *, … Continue reading

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Book Review: Topology, James R. Munkres

Topology (2nd Edition) This book is the best introductory book on Topology, a Maths module taken in university. I have wrote a short book review on it. Excerpt: Book Review: Topology Book’s Author: James R. Munkres Title: Topology Prentice Hall, … Continue reading

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