Tag Archives: Topology

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 →

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

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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)

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →

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 →