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Category# Algebraic topology

Summary

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
Below are some of the main areas studied in algebraic topology:
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
Homology
In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
Cohomology
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory.
Manifold
A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions.

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Homotopy category of chain complexes

In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.

Real projective space

In mathematics, real projective space, denoted \mathbb{RP}^n or \mathbb{P}_n(\R), is the topological space of lines passing through the origin 0 in the real space \R^{n+1}. It is a compact, smooth manifold of dimension n, and is a special case \mathbf{Gr}(1, \R^{n+1}) of a Grassmannian space. As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ∖ under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1 ∖ one can always find a λ such that λx has norm 1.

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.

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Abstract algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.

Algebraic geometry

Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations.

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

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Collapsing cell complexes was first introduced in the 1930's as a way to deform a space into a topological-equivalent subspace with a sequence of elementary moves. Recently, discrete Morse theory techniques provided an efficient way to construct deformation retracts collapsing one space into the other while preserving global topological properties. This type of collapse, called a Morse matching, has been widely used to speed up computations in (persistent) homology by reducing the size of complexes. Unlike classical collapses, in this thesis we consider topological spaces equipped with signals or directions. The main goal is then to reduce the size of the spaces while preserving as much as possible of both the topological structure and the properties of the signals or the directions. In the first part of the thesis we explore collapsing in topological signal processing. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resultant Hodge decomposition.In Article 2.1 we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. We first prove that any deformation retract of real finite-dimensional based chain complex is equivalent to a Morse matching. We then study the interaction between the Hodge decomposition and signal compression and reconstruction. Specifically, we prove that parts of a signal's Hodge decomposition are preserved under compression and reconstruction for specific classes of discrete Morse deformation retracts of a given based chain complex. Finally, we provide an algorithm to compute Morse matchings with minimal reconstruction error.Complementary to our theoretic results in topological signal processing, we provide two applications in this field. Article 2.2 extends graph convolutional neural networks to simplicial complexes, while Article 2.3 presents a novel algorithm, inspired by the well-known spectral clustering algorithm, to embed simplices in a Euclidean space. The object of our studies in the second part of the thesis is topological spaces equipped with a sense of direction. In the directed setting, the topology of the space is characterized by directed paths between fixed initial and terminal points. Motivated by applications in concurrent programs, we focus on directed Euclidean cubical complexes and their spaces of directed paths.In Article 3.1 we define a notion of directed collapsibility for Euclidean cubical complexes using the notion of past links, a combinatorial local representations of cubical complexes. We show that this notion of collapsability preserves given properties of the directed path spaces. In particular, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex.In Article 3.2 we extend these results, providing further conditions for directed collapses to preserve the contractability or conneectdness of spaces of directed paths. Furthermore, we provide simple combinatorial conditions for preserving the topology of past links. These conditions are the first step towards developing an algorithm that checks at each iteration if a collapse preserves certain properties of the directed space.

The field of computational topology has developed many powerful tools to describe the shape of data, offering an alternative point of view from classical statistics. This results in a variety of complex structures that are not always directly amenable for machine learning tasks. We develop theory and algorithms to produce computable representations of simplicial or cell complexes, potentially equipped with additional information such as signals and multifiltrations. The common goal of the topics discussed in this thesis is to find reduced representations of these often high dimensional and complex structures to better visualize, transform or formulate theoretical results about them. We extend the well known graph learning algorithm node2vec to simplicial complexes, a higher dimensional analogue of graphs. To this end we propose a way to define random walks on simplicial complexes, which we then use to design an extension of node2vec called k-simplex2vec, producing a representation of the simplices in a Euclidean space. Furthermore, the study of this method leads to interesting questions about robustness of graph and simplicial learning methods. In the case of graphs, we study node2vec embeddings arising from different parameter sets, analysing their quality and stability using various measures. In the topic of signal processing, we explore how discrete Morse theory can be used for compression and reconstruction of cell complexes equipped with signals. In particular we study the effect of the compression of a complex on the Hodge decomposition of its signals. We study how the signal changes through compression and reconstruction by introducing a topological reconstruction error, showing in particular that part of the Hodge decomposition is preserved. Moreover, we prove that any deformation retract over R can be expressed as a Morse deformation retract in a well-chosen basis, thus extending the reconstruction results to any deformation retract. In addition, we introduce an algorithm to minimize the loss induced by the reconstruction of a compressed signal. Finally, we use discrete Morse theory to compute an invariant of multi-parameter persistent homology, the rank invariant. We can restrict a multi-parameter persistence module to a one- dimensional persistence module along any line of positive slope and compute the one-dimensional analogue of the rank invariant, namely the barcode. Through a discrete Morse matching we can determine critical values in the multifiltration, which in turn allows us to identify equivalence classes of lines in the parameter space. In our main result, we explain how to compute the barcode along any given line of an equivalence class given the barcode along a representative line. This provides a way to fiber the rank invariant according to the critical values of a discrete Morse matching and to perform computations in the corresponding one-dimensional module, which is much better understood.

In this thesis, we investigate the inverse problem of trees and barcodes from a combinatorial, geometric, probabilistic and statistical point of view.Computing the persistent homology of a merge tree yields a barcode B. Reconstructing a tree from B involves gluing the branches back together. We are able to define combinatorial equivalence classes of merge trees and barcodes that allow us to completely solve this inverse problem. A barcode can be associated with an element in the symmetric group, and the number of trees with the same barcode, the tree realization number, depends only on the permutation type. We compare these combinatorial definitions of barcodes and trees to those of phylogenetic trees, thus describing the subtle differences between these spaces. The result is a clear combinatorial distinction between the phylogenetic tree space and the merge tree space.The representation of a barcode by a permutation not only gives a formula for the tree realization number, but also opens the door to deeper connections between inverse problems in topological data analysis, group theory, and combinatorics.Based on the combinatorial classes of barcodes, we construct a stratification of the barcode space. We define coordinates that partition the space of barcodes into regions indexed by the averages and the standard deviations of birth and death times and by the permutation type of a barcode. By associating to a barcode the coordinates of its region, we define a new invariant of barcodes.These equivalence classes define a stratification of the space of barcodes with n bars where the strata are indexed by the symmetric group on n letters and its parabolic subgroups.We study the realization numbers computed from barcodes with uniform permutation type (i.e., drawn from the uniform distribution on the symmetric group) and establish a fundamental null hypothesis for this invariant. We show that the tree realization number can be used as a statistic to distinguish distributions of trees by comparing neuronal trees to random barcode distributions.