Summary Algebraic Topology for Data Scientists arxiv.org
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"Algebraic Topology for Data Scientists" is a comprehensive textbook that teaches algebraic topology concepts, including point-set topology, abstract algebra, and traditional homology theory, specifically tailored for data science applications.
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Slide Presentation (10 slides)
Key Points
- Algebraic topology is used in data science for analyzing time series data and classifying internet of things (IoT) data.
- Topological data analysis (TDA) is introduced as a methodology in data science that utilizes algebraic topology.
- Dimensionality reduction techniques such as Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP) are applied in the context of algebraic topology for data analysis.
- The notation f : (X, A) ? (Y, B) represents a generalization of a triple in algebraic topology.
- Homotopy classes of algebraically trivial maps from a 3-dimensional space X into S2 correspond to elements of H.
- The Adem relation is used in Example 11.4.2 to demonstrate the equation Sq 2 Sq 4 = Sq 6 + Sq 5 Sq 1.
- Example 11.4.3 proves the coefficient of x in a certain equation using algebraic topology principles.
Summaries
23 word summary
"Algebraic Topology for Data Scientists" is a textbook introducing algebraic topology in data science, covering point-set topology, abstract algebra, and traditional homology theory.
38 word summary
"Algebraic Topology for Data Scientists" is a textbook that introduces the use of algebraic topology in data science. It covers topics such as point-set topology, abstract algebra background, and traditional homology theory. The text discusses the application of
212 word summary
"Algebraic Topology for Data Scientists" by Michael S. Postol, Ph.D. provides an introduction to algebraic topology and its applications in data science. The document covers point-set topology, abstract algebra background, traditional homology theory,
This textbook introduces topological data analysis (TDA) and its applications. It covers various topics in algebraic topology and their applications to data science, including multivariate time series, classification of internet of things data, Q-analysis, sensor coverage, the
Algebraic Topology for Data Scientists: A concise summary of key points and details.
Paragraph 1: The excerpt discusses the use of algebraic topology in data analysis, specifically in analyzing time series data and classifying internet of things (IoT
The excerpt discusses the use of algebraic topology in data science, specifically in the context of dimensionality reduction techniques such as Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). It explains that UMAP
The notation f : (X, A) ? (Y, B) is used to denote a generalization of a triple. Homotopy classes of algebraically trivial maps from a 3-dimensional space X into S2 correspond to elements of H
In Example 11.4.2, the Adem relation is used to show that Sq 2 Sq 4 = Sq 6 + Sq 5 Sq 1. Example 11.4.3 proves that the coefficient of x in
5101 word summary
Algebraic Topology for Data Scientists by Michael S. Postol, Ph.D. at The MITRE Corporation. Acknowledgements to collaborators and contributors. Introduction and Point-Set Topology Background.
Algebraic Topology for Data Scientists summarized.
Abstract Algebra Background: Groups, Exact Sequences, Rings and Fields, Vector Spaces, Modules, and Algebras, Category Theory. Traditional Homology Theory: Simplicial Homology, Simplicial Complexes, Homology Groups,
Homomorphisms, topological invariance, Eilenberg Steenrod Axioms, long exact sequences, Mayer-Vietoris sequences, singular and cellular homology.
Summary: The document discusses CW complexes, homology of CW complexes, and projective spaces. It also covers persistent homology, computational issues, bar codes, persistence diagrams, persistence landscapes, zig-zag persistence, and distance measures.
5.6 Sublevel Sets, 5.7 Graphs, 5.7.1 Review of Graph Terminology, 5.7.2 Graph Distance Measures, 5.7.3 Simplicial Complexes from Graph
5.9.2 SEQUITUR, 5.9.3 Representative Pattern Mining, 5.9.4 Converting Multivariate Time Series to the Univariate Case or Predicting Economic Collapse, 5.9.5 Classification of
Construction, implementation, height function examples, breast cancer research, simplicial sets, ordered simplicial complexes, delta sets, definition of simplicial sets, geometric realization, UMAP, theoretical basis, computational view, weaknesses of UMAP,
Algebraic Topology for Data Scientists summarized.
Unfinished ideas: time series of graphs, directed simplicial complexes, computer intrusion detection, market basket analysis. Introduction to cohomology: homological algebra, tensor product, Ext, Tor.
Definition of Cohomology, Cup Products, Universal Coefficient Theorems, Homology and Cohomology of Product Spaces, Ring Structure of the Cohomology of a Product Space, Persistent Cohomology, Ripser.
Homotopy Theory: Extension and Lifting Problems, Fundamental Group, Fiber Bundles, Hopf Maps, Paths and Loops, Higher Homotopy Groups.
Definition of higher homotopy groups, relative homotopy groups, boundary operator, induced homomorphisms, properties of homotopy groups, homotopy systems vs. Eilenberg-Steenrod axioms, operation of the fundamental group
Kenzo, classical obstruction theory, extension problem, obstruction cochain, difference cochain, Eilenberg's extension theorem, obstruction sets, application to data science, possible application to image classification, simplicial sets.
Algebraic Topology for Data Scientists, important details, key points.
Algebraic Topology for Data Scientists: Summary of Key Points - Section 10.3.2 discusses Kan Complexes. - Section 10.4 explores the computability of the Extension Problem. - Section 11 focuses on Steen
The Cartan Formula, Cartesian Products, Adem Relations, Hopf Invariant, Steenrod Algebra, Cohomology of Eilenberg-Mac Lane Spaces, Bocksstein Exact Couple, Serre's Exact Sequence, Transgression, Coh
Vector Bundles, Stiefel-Whitney Classes, Grassmann Manifolds, Steenrod Squares, Homotopy Groups of Spheres, Classes of Abelian Groups, Finiteness of Homotopy Groups, Iterated
Conclusion.
Algebraic Topology for Data Scientists summarized.
List of Tables: Group Laws, Results for TDA Multivariate Case, Social Amenities at the University of Essex, Stable Homotopy Groups ? i S for i ? 19, Homotopy Groups ? i (S n ) for
Klein Bottle, Boy's Surface, Suspension of a Circle, Topologist's Sine Curve. Spaceship Earth, Convex vs Non-Convex Set, Good and Bad Simplical Complexes, Boundary Examples, Homology Examples.
Excision Example, Barycentric Subdivision, Stereographic Projection, Zig-zag Lemma, Non-triangulable CW Complex, Torus and Klein Bottle as CW Complexes.
Ring of Data Points, Vietoris-Rips Complexes, Bar Codes and Persistence Diagrams, Persistence Diagrams vs. Persistence Landscapes, Example Matching of Two Persistence Diagrams, Sublevel Sets of the Height Function, Barcode and Persistence Diagram of
Algebraic Topology for Data Scientists: Summary of Figures.
Pipeline for persistent images, criterion for relative compactness, tent function. A generator of H2(Rs, Fs) killed by inclusion. Simplicial complexes for examples. Face maps, I?I glued into a cone, delta set,
Degenerate singular simplex, authorized user and lion impostor, example of a process graph, example of a window graph, confusion matrix, S1 ? S1 ? S2, first quadrant spectral sequence, Kenzo the Cat, maps involved in a
This textbook introduces topological data analysis (TDA) and its applications. It provides necessary background in point-set topology, abstract algebra, and homology theory. TDA is a good tool to be used alongside machine learning methods. Algebraic topology is
The text discusses various topics in algebraic topology and their applications to data science. It covers multivariate time series, classification of internet of things data, Q-analysis, sensor coverage, the "mapper" algorithm, simplicial sets, UMAP,
T 0: distinct points can be separated by open sets T 1: distinct points can be separated by open sets containing each point T 2: distinct points can be separated by disjoint open sets T 3: distinct points can
Each subset of X is the intersection of the open sets containing it. A T1 space is closed and a union of open sets, making it closed. A is the intersection of open sets, so it is open. If (3) holds,
Groups are sets with a binary operation that follows specific rules. They can be finite or infinite, and can be abelian or non-abelian. Examples include the integers, real numbers, permutation groups, and cyclic groups. Subgroups are subsets of
H inherits the binary operation from G, being additive or multiplicative based on G. H is assumed to be abelian, but non-abelian groups are discussed later. H needs closure, identity, and inverses to be a group. Sub
Short exact sequences and their properties in algebraic topology.
This chapter introduces simplicial homology, which is a homology theory commonly used in topological data analysis. It begins by defining simplicial complexes, which are built from simplices. A simplex is a geometrically independent set of points in
The excerpt discusses barycentric coordinates, simplicial complexes, and homology theory. It explains the concepts of simplices, polytopes, and boundaries. It also introduces the idea of relative homology and the excision theorem. The
The homology of objects changes when g = ?g and 2g = 0. Homology with integer coefficients can be used to find it for any other coefficient group using the Universal Coefficient Theorem. Cohomology groups do not provide
Algebraic Topology for Data Scientists: Summary
Traditional homology theory outline of proof. Step 1: Group of p-cycles and p-boundaries. Wp is the group of weak boundaries. Step 2: Choose basis for Cp
Chain homotopy between simplicial maps implies equal induced homomorphisms for homology. Contiguous simplicial maps have a chain homotopy and equal homology. Homeomorphic spaces have the same homotopy groups and homology
The chain complex C(K) consists of chain groups of K and boundaries. Homomorphisms induced by h have nice properties. Topological invariance and relative homology hold. Homotopy equivalence and homotopy classes are defined. Hom
The unit sphere is a deformation retract of punctured Euclidean space. Euclidean spaces of different dimensions are not homeomorphic. The point at infinity can be added to Rn to produce the sphere. Stereographic projection maps the sphere with the North
Homotopy maps give rise to the same homomorphisms in homology. The composite of two maps is the zero homomorphism. There is no retraction from B n+1 to S n. The Brouwer Fixed-Point
The relationship between Euler characteristic and genus is explained. Eilenberg-Steenrod axioms for homology theory are introduced. The long exact homology sequence of a pair is discussed. The zig-zag lemma is presented as a generalization of the
Mayer-Vietoris sequences compute homology, utilizing exact sequences. A short exact sequence of chain complexes is constructed using the zig-zag lemma. The homology of the middle chain complex is determined. Cones are acyclic and plug up holes
Axioms of traditional homology theory, compact support, and dimension. Singular and cellular homology.
Spaces are represented as CW complexes, which consist of cells that are homeomorphic to open balls. For example, a 0-cell is a point and a 1-cell is a line segment. Spaceship Earth at Disneyworld can be represented as a
A non-triangulable CW complex is described using a square and triangle. The construction of the space is explained, leading to a contradiction. Two theorems about CW complexes are stated. The process of computing the homology of a CW complex
Assuming X is triangulable, let K be the complex that triangulates X. Open p-cells in K are polytopes of subcomplexes. The group H_p(e, e') is the group of p-chains carried by
The n-skeleton is Pn and there is an analogous complex projective space. The complex n-sphere corresponds to S2n+1. CPn is a CW complex with one cell in each dimension 2j for 0 ≤ j
In algebraic topology, we can use persistent homology to analyze the shape of data. By growing points into circles of increasing radius, we can identify holes and describe the shape. Visualizations such as bar codes, persistence diagrams, and persistence landscapes display
The C?ech complex and the Vietoris-Rips complex are two methods used in algebraic topology to analyze data. The C?ech complex is an abstract simplicial complex that represents the union of balls centered at data points. The Vietoris
In this case, the degree of f is deg f = n. Now let f (x) = a 0 + a 1 x + a 2 x 2 + . . . + a n x n and g(x) =
Persistence complex and persistence module are of finite type. Graded ring/module is defined. Persistence module corresponds to non-negatively graded module over R[t]. P-intervals represent birth and death times of homology classes. Bar codes, persistence diagrams,
Algebraic Topology for Data Scientists: A concise summary of key points and details.
1. A distance measure between bar codes for different point clouds is important in algebraic topology for data scientists. 2. Persistence diagrams provide a more elegant visualization
This theorem allows us to produce barcodes in this case by writing one bar extending from a i to b i for each indecomposable zigzag submodule. The Hausdorff distance between A and B is d H (A, B) =
Persistence landscapes provide statistical results like the law of large numbers and central limit theorem. A mean persistence landscape exists and landscapes can be measured as a Banach space. Rn is a normed space. A Banach space is complete and every closed subs
TDA has applications in cancer research and finance. Free software available. Manifolds, Morse theory, and sublevel sets. Examples of sublevel sets on a torus and surfaces. Use of persistence diagrams for detecting changes in elephant population and grayscale
A graph consists of vertices and edges, with adjacent vertices called neighbors. A graph can be represented by an adjacency or incidence matrix. The degree of a vertex is the number of its neighbors. A subgraph is a subset of a graph, while a
A digraph's adjacency matrix is defined as a matrix representing the number of edges between vertices. The outdegree of a vertex is the number of edges with that vertex as the tail, and the indegree is the number of edges with that vertex as
A partially ordered set can be represented by a collection of sets. There are five types of simplicial complexes that can be built from a graph. Machine learning algorithms can utilize features such as the dimension of the complex, number of simplices in each
The author used SAX for time series analysis and built a tool for classification and anomaly detection. TDA was crucial for dealing with multivariate time series. They collaborated with George Mason University and developed a software tool called TSAT. The SAX algorithm replaces real
Every rule is used more than once. Examples show rule utility. Constraints can be violated. SEQUITUR algorithm enforces constraints. Representative Pattern Mining algorithm classifies time series. SAX and SEQUITUR used in preprocessing. Multivariate time series conversion using
The excerpt discusses the use of algebraic topology in data analysis. It explores the application of topology to time series data and classification of internet of things (IoT) devices. The first part of the excerpt focuses on using persistence landscapes to analyze time series
Advantage of smaller time windows and simpler simplical complex [34]. Construct filtration of complexes using window size k [34]. Convert resulting diagram to persistence image [34]. Classify images using convolutional neural net [34]. Results showed high accuracy,
The summary includes key points from the original text and is organized into separate paragraphs for readability.
Paragraph 1: The persistence diagram of a graph is compared to an unweighted ordinal partition graph. The network periodicity score measures the difference between the two diagrams
Template functions turn persistence diagrams into vectors for classification. Two types of functions are used: tent functions and interpolating polynomials. Teaspoon package can compute both. A persistence diagram represents homology classes in R2. The dimension is fixed. Multip
Let S = (D n In ? N ), D n = ((n, n + 1)) with ? D n (n, n + 1) = 1. This is bounded and UODF but not ODBB.
The text explains a theorem characterizing elements of the dual of the dual. It introduces a template system for a locally convex space and describes how template functions can approximate continuous functions on persistence diagrams. The text then presents two families of template functions: tent functions
PERSISTENT HOMOLOGY intro, software options, and TDA applications discussed. Q-Analysis and sensor coverage explored. Mapper, simplicial sets, and UMAP algorithm covered. Q-Analysis explained using topological representation and q-
There are 4 simplices of dimension at least 2, Q2 = 4. Q1 = 4, Q0 = 1. The structure vector is Q = [1, 4, 4, 1]. The
Theorem 6.1.1 states that two simplices in a complex share a t-face if and only if their monomial representations satisfy certain conditions. Q-nearness is introduced as a concept of nearness between simplices. Chains of
Chapter 9 provides more details on the shomotopy version of algebraic topology. The University of Essex study explores how this applies to the university's structures, including physical, administrative, committee, and social structures. The report covers various levels and
The problem of sensor coverage is solved using algebraic topology. A collection of graphs is built based on the nodes and their communication range. The sensor cover is determined by the homomorphism of the relative homology groups. The Cech complex is difficult
In [116], Nicolau, Levine, and Carlsson used Mapper to discover a new type of breast cancer cell that had a 100% survival rate and no metastasis. Mapper software is available in open source versions in both R and Python,
If the filter function is real valued, our complexes are 1-dimensional graphs. To look at a higher dimensional simplical complex, we need Rm for m > 1 as our parameter space. We can cover R2 with overlapping rectangles and add
This group had 100% survival rate with no recurrence of disease. Simplicial sets are a key part of UMAP and combinatorial homotopy theory. Simplicial maps are completely determined by their action on the vertices.
To understand the UMAP paper, it is important to understand the category ?. This category has sets [n] as objects and strictly order preserving functions as morphisms. The dual category ?op goes in the opposite direction. A Delta set is a cov
Simplicial sets are contravariant functors in algebraic topology. Simplicial sets can be thought of as objects in a category with morphisms that are functions between sets. The realization of a simplicial set corresponds to a
The UMAP algorithm is a dimensionality reduction technique used in machine learning. It preserves the manifold structure of high dimensional data. The theoretical basis of UMAP involves concepts from algebraic topology, differential geometry, and category theory. The algorithm approximates a
Geodesic distance between points on a manifold can be normalized. Simplicial sets and colimits are used to combine local metric spaces. Fuzzy set theory represents varying degrees of membership. Sheaves in algebraic geometry can represent fuzzy sets.
A full functor is surjective, a faithful functor is injective. A fully faithful functor is both. The category of fuzzy simplicial sets is a subcategory of sheaves. Fuzzy simplicial sets are related to metric spaces. An
UMAP is a graph learning algorithm that constructs a weighted k-neighbor graph to preserve the topological structure of the data. It uses a force directed graph layout algorithm for the graph layout step. UMAP should not be used if interpretability of reduced
The excerpt discusses the use of algebraic topology in data science, specifically in the context of dimensionality reduction techniques such as Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). The SNE algorithm involves
Algebraic topology is mainly used for undirected graphs. Directed graphs can be applied to various problems. The directed flag complex is a key structure used in the Blue Brain Project. Chains and boundaries are used to compute homology. FLAGSER is a
Using algebraic topology, we can add features related to graph complexes and create directed flag complexes. These complexes can be used to classify and cluster customers based on their transactions. The Apriori Algorithm efficiently finds association rules with high support and confidence. We
Section 8.3 discusses the cup product for cohomology. Section 8.4 covers Universal Coefficient Theorems. Section 8.5 discusses homology and cohomology of product spaces. Section 8.6 focuses on
The text discusses exact sequences, homomorphisms, and tensor products in algebraic topology. It provides formulas for determining Hom(A, B) when A and B are finitely generated abelian groups. It also defines the tensor product and its
Proof idea: Let G be a subgroup of B?B? generated by elements of the form b?b? in the theorem. Map G to zero, inducing a homomorphism ?:(B?B?)/G?C?C?.
Ext(A, B) is a functor that takes two abelian groups and returns a third abelian group. It is contravariant in the first variable and covariant in the second. A free resolution of an abelian group A is a short
1. There are natural isomorphisms between certain torsion subgroups. 2. Torsion-free modules have zero torsion. 3. There is an exact sequence involving torsion subgroups. 4. The exact sequence implies certain results for
If K is connected, then H? 0 (K; G) = 0. In general, H 0 (K; G) ? = H? 0 (K; G) ? G . The group of relative cochains
The homology and cohomology groups are different for the torus and the Klein bottle. If two spaces have the same homology groups, they have the same cohomology groups. Cohomology has a ring structure that homology lacks.
The wedge product of topological spaces X and Y is the union of X and Y with one point in common. The figure 8 is an example of the wedge product. A donut and a torus are not homeomorphic or homotopy
Homology and cohomology groups of torus and Klein bottle. UCT for homology. Homology and cohomology of product spaces. Ku?nneth Theorem.
The Ku?nneth Theorem for Chain Complexes states that there is an exact sequence involving homomorphisms induced by chain maps. The sequence splits but not naturally. The Ku?nneth Theorem also implies that the homology of the
Ku?nneth Theorem states exact sequence for topological spaces. Homology cross product defined. Cochain complex and tensor product explained. Cohomology cross product defined. Ku?nneth Theorem for Cohomology stated. Ring structure
Theorem 8.6.3 states an equation involving the interchange of coordinates. Theorem 8.6.4 discusses properties of the cohomology rings of products. Theorem 8.6.5 relates the cross product to rings
Algorithm for determining if a class in a simplicial complex is decomposable. Ripser is the fastest software for computing persistent homology barcodes. It uses optimizations such as clearing inessential columns and computing cohomology. Implicit matrix reduction is also
Persistence intervals in topological data analysis only use a small fraction of algebraic topology. Homotopy theory is closely related to homology theory. Homotopy groups are easier to calculate than homology groups. Homotopy is important in obstruction theory
The lifting problem involves finding a map that satisfies certain conditions. The fundamental group classifies maps between S1 and a space X. The exponential map is a homomorphism. Paths and loops are defined. The covering path property and covering homotopy
The fundamental group of a space X with base point x_0 is denoted by π_1(X, x_0) and is the (non-abelian) free group generated by loops. If X is pathwise connected, we can
Let E = B x D with projection p: E ? B. A map p: E ? B is a fibering if it has the PCHP. A map p: E ? B has the bundle property if there exists a space D
Coordinate bundles and their properties are defined. Equivalence of coordinate bundles is discussed. Cross sections in bundle spaces are defined. The existence of local cross sections and the conditions for global cross sections are explained. The concept of fiber maps and induced maps is introduced
The first Hopf bundle is a fiber bundle that identifies antipodal points on a sphere. The complex analogue is a fiber space that consists of the unit sphere in C and the quotient space of the unit sphere under an equivalence relation. This bundle has
The text discusses the use of quaternions to represent rotations in R3. It explains that quaternions are associative but not commutative, and that rotations can be defined using unit quaternions. The text also introduces the Hopf map
The notation f : (X, A) ? (Y, B) is used to denote a generalization of a triple. Let n be greater than 1 and I n be the n-dimensional cube that is the product of I = [0
The homotopy classes of algebraically trivial maps from a 3-dimensional space X into S2 correspond to elements of H3(X). For S3, the homotopy classes of maps to S2 correspond to integers. Relative homotopy
Homotopy groups have properties similar to homology groups. Homotopy sequences of triplets are exact. Homotopic maps have equal homotopy transformations. Homotopy equivalence induces isomorphisms. Fiber spaces have a long exact sequence
Path components in a space have the same zeroth homotopy group. For n > 0, every path connecting two points in a space gives an isomorphism between the nth homotopy groups. The fundamental group acts on the higher hom
The product of two spaces is easier for homotopy than for homology. The formula for homotopy is simpler. The one point union of spaces is more complicated for homotopy than for homology. The Hurewicz Theorem
The first nonzero homotopy group is isomorphic to the homology group. Excision does not generally hold for homotopy, but it does hold in the stable range of dimensions. The suspension map is important for later constructions. The Freundenthal
For n > 1 and ? an abelian group, there exists a CW complex with the homotopy type of a K(?, n) space. The dimension of this space is at most n+1 and it is (n-1
Two spaces have the same n-homotopy type if there are maps between them that satisfy certain conditions. Two CW complexes have the same n-type if their n-skeletons have the same (n-1)-homotopy type. The
The Leray-Serre spectral sequence is an important tool in algebraic topology. It can be used to compute homotopy groups of spheres and is described in books by McCleary, Mac Lane, Mosher and Tangora. The Leray
The Leray-Serre spectral sequence is used to compute homology of fiber spaces. It relates the homology of the base and the fiber to that of the total space. The Wang sequence and the Gysin sequence are examples of spectral sequences.
A bounded filtration of a differential graded module A determines a spectral sequence (E r , d r ) that converges to H(A). The associated spectral sequence derived from a point cloud has not been fully explored. An exact couple C = (D,
There exists a non-trivial ultranet, but it has not been constructed. The existence of ultrafilters implies the existence of nontrivial ultranets. Constructive mathematics distinguishes between the converse of a proposition being false and the proposition being
Relations satisfied: f g = id c, gf + dh + hd = id C?, f h = hg = hh = 0. Theorem 9.9.10 states that a reduction ? : C? ? C is equivalent to a
In the first section, obstruction theory and its relation to the extension problem are discussed. A data science application involving the Hopf map is presented. Additional simplical set concepts needed for computational papers are discussed. Obstruction theory is used for constructing cross sections
If n = 1, then h is an epimorphism and the kernel of h is contained in the kernel of k. A well-defined homomorphism ? = k ? h ?1 : Z n (B) ? ? n (Y
Homotopic maps have the same (n + 1)-dimensional obstruction set. A proper cellular map induces a homomorphism that sends O n+1 (f) into O n+1 (f?). If (K, L)
The excerpt discusses the use of obstruction theory in image segmentation and classification. It suggests considering the distance between pixels in an image and varying inter-pixel distances to make things more interesting. The excerpt also mentions representing a moving picture by stacking frames and using distances
The text discusses simplicial sets and their properties, particularly in relation to algebraic topology. It mentions the degeneracy and nondegeneracy of simplices, as well as the computation of face maps. The concept of Kan complexes and the
There is an algorithm to compute the abelian group of homotopy classes of simplicial maps from X to Y. The algorithm runs in polynomial time and expresses the group as a direct product of cyclic groups. The algorithm also determines the element of
The P i have an H-space structure, outlined in Section 5 of [26]. The group structure on the P i induces group operations ? i? and ? i? on the set SM ap(X, P i ). The semi-effective representation of
Steenrod squares used for classification of vector bundles. Historical ties to obstruction theory and extension problem. Steenrod's original motivation: generalizing map classification problem. Cup-i product and coboundary formula. Homotopy classification theorem and extension problem.
A mapping f : X ? Y is called an extension of h if f (x) = h(x) for each x ? A. In terms of diagrams, we have the following: X g A f Y h. We use algebraic topology
In this case, im f ? is a subring. A cohomology operation Sq 2: H 3 (X; Z 2 ) ? H 5 (X; Z 2 ) shows that the retraction cannot exist.
Cohomology of K(Z2,1) space is examined. The cell structure of S? is acyclic. The homology of P? is Z2 in odd dimensions and 0 in even dimensions. The ring structure of P?
The cup-i products are derived using the Acyclic Carrier Theorem. The carrier C is acyclic and h-equivariant. The map ? is used for the cup-i products. Sq i is a (group) homomorphism. Sq i is
H ? (P 2 ; Z 2 ) is a truncated polynomial ring generated by u, a nonzero class in H 1 (P 2 ; Z 2 ). The operation Sq 1 is nontrivial in H 2 (
In algebraic topology, the Ku?nneth Theorem states that the polynomial ring H ? (K n ; Z 2 ) is generated by x 1 , . . . , x n , where x i is the nonzero one dimensional class
In Example 11.4.2, it is shown that Sq 2 Sq 4 = Sq 6 + Sq 5 Sq 1 by using the Adem relation. In Example 11.4.3, it is proven that
High school algebra students will be happy to know that the coefficient of x in the expansion (1 + x) b is a b. Mosher and Tangora's main result states that if there exists a map f: S 2n-1
An augmented graded R-algebra is connected if it is an isomorphism. The tensor algebra ?(M) is defined as a graded R-algebra. The Steenrod algebra A is a quotient algebra of ?(M)/Q. The
The cohomology ring is an algebra over an algebra. The cohomology ring with Z2 coefficients is an algebra over the Hopf algebra A. The structure of the cohomology ring H?(Z2, n, Z2) is
The exact sequence for a fiber space is given by Ei,j where i + j = n. The transgression map corresponds to dn,n,0 and is only defined if n < p + q. Serre's cohomology exact sequence for
Theorem 11.7.9 states that H?(Z2, n; Z2) is a polynomial ring over Z2 with generators (SqI(in)) for admissible sequences with excess less than n. Theorem 11.7
Steenrod squares, reduced powers, and the Bockstein homomorphism are discussed. The Steenrod algebra A(p) is defined as a graded associative algebra generated by elements P i and ? subject to certain relations. Admissible monom
The bundle ? n 1 over P n is not trivial for n ? 1. A cross-section of a vector bundle ? with base space B is a continuous function s : B ? E(?) which takes b ? B into the corresponding fiber
Stiefel-Whitney classes classify vector bundles like homology, cohomology, and homotopy. Trivial vector bundles have Stiefel-Whitney classes of 0. Grassmann manifolds have cohomology
The formula for D i in algebraic topology is derived using the Elilenberg-Zilber contraction. The homotopy operator is defined in terms of partitions and shuffles of face and degeneracy operators. The complexity of evaluating an element of
The formulas for computing Steenrod squares are reduced to eliminate terms with degeneracy operators. Explicit formulas are provided for the Eilenberg-Zilber contraction and cup-i product. The complexity of computing Steenrod squares is evaluated and shown to
The stable homotopy group sequence is discussed, with the suspension map being an isomorphism for certain dimensions. The stable range and unstable range are defined, and the groups ?iS for i ? 19 are listed. The finite nature of
Classes of abelian groups closed under subgroups, quotient groups, and group extensions. Homomorphisms can be C-monomorphism, C-epimorphism, or C-isomorphism. Axioms 1, 2A,
6. f # : ? i (A) ? ? i (X) is a C p -isomorphism for i < n and a C p -epimorphism for i = n. 7. ? i (A) and ?
The p-primary components of various spaces are isomorphic under certain conditions. The p-primary component of ? m (S 3 ) is 0 if m < 2p and Z p if m = 2p. Pseudo-projective spaces
Theorem 12.3.16: ? n+1 (S n ) ? = Z 2 for n ? 3. Theorem 12.3.17: ? n+2 (S n ) ? = Z 2
Isaksen et al. used the motivic homotopy theory of Morel and Voevodsky to compute stable homotopy groups up to i=90, beating previous results. Computer calculations of Ext groups were used, producing data up
Blue Brain Project, homology and shellability, time series analysis, random forests, finite computability, large ext modules, the cohomology of the mod 2 Steenrod algebra, statistical topological data analysis, computing all maps into a
Algebraic Topology for Data Scientists: Relevant research papers and textbooks.
Algebraic Topology for Data Scientists: important references on the topic.
The text includes various citations and references to books, articles, and software related to algebraic topology, data analysis, and mathematical concepts.
Approximating continuous functions on persistence diagrams using template functions. Classification of mappings of the 3-dimensional complex into the 2-dimensional sphere. Investigations in homotopy theory of continuous mappings. Homotopy and topological actions on spaces with few homot
The bibliography includes various mathematical texts and resources.