# Quantitative Homotopy Theory in Topological Data Analysis

## Related Articles

- Using non-cofinite resolutions in shape theory. Application to Cartesian products. Mardešić, Sibe // Mathematical Communications;2011, Vol. 16 Issue 2, p301
The strong shape category of topological spaces SSh can be defined using the coherent homotopy category CH, whose objects are inverse systems consisting of topological spaces, indexed by cofinite directed sets. In particular, if X, Y are spaces and q : Y â†’ Y is a cofinite HPol-resolution...

- RATIONAL HOMOTOPY THEORY OF MAPPING SPACES VIA LIE THEORY FOR Lâˆž-ALGEBRAS. BERGLUND, ALEXANDER // Homology, Homotopy & Applications;2015, Vol. 17 Issue 2, p343
We calculate the higher homotopy groups of the Deligneâ€“ Getzler âˆž-groupoid associated to a nilpotent Lâˆž-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

- A TOPOLOGICAL FIBREWISE FUNDAMENTAL GROUPOID DAVID MICHAEL ROBERTS. ROBERTS, DAVID MICHAEL // Homology, Homotopy & Applications;2015, Vol. 17 Issue 2, p37
It is well known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological...

- Examples of Rational Toral Rank Complex. Yamaguchi, Toshihiro // International Journal of Mathematics & Mathematical Sciences;2012, p1
There is a CW complex Î¤(X), which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of X associated with rational toral ranks and also presents certain relations in them. We call it the rational toral rank complex of X. It...

- GENERALIZED DAVIS-JANUSZKIEWICZ SPACES MULTICOMPLEXES AND MONOMIAL RINGS. TREVISAN, ALVISE J. // Homology, Homotopy & Applications;2011, Vol. 13 Issue 1, p205
We show that every monomial ring can be realized topo-logically by a certain topological space. This space is called a generalized Davis-Januszkiewicz space and can be thought of as a colimit over a multicomplex, a combinatorial object generalizing a simplicial complex. Furthermore, we show that...

- SPACES OF ALGEBRAIC AND CONTINUOUS MAPS BETWEEN REAL ALGEBRAIC VARIETIES. Adamaszek, Michal; Kozlowski, Andrzej; Yamaguchi, Kohhei // Quarterly Journal of Mathematics;Dec2011, Vol. 62 Issue 4, p771
We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset...

- Chapter 10: Covering spaces, covering groupoids: 10.1 Covering maps and covering homotopies. Brown, Ronald // Topology & Groupoids;2006, p359
Chapter 10.1 of the book "Topology and Groupoids" by Ronald Brown is presented. It focuses on the concept of covering maps and covering homotopies. It defines a map of topological spaces which is called a covering map. Moreover, it proves the path lifting property with the use of the covering...

- STABLE MAPS BETWEEN 4-MANIFOLDS AND ELIMINATION OF THEIR SINGULARITIES. SAEKI, OSAMU; SAKUMA, KAZUHIRO // Journal of the London Mathematical Society;06/01/1999, Vol. 59 Issue 3, p1117
Let f:Mâ€™N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:Mâ€™N homotopic to f which has only fold and cusp singularities. This together with...

- Polynomial-Time Homology for Simplicial Eilenberg-MacLane Spaces. Krčál, Marek; Matoušek, Jiří; Sergeraert, Francis // Foundations of Computational Mathematics;Dec2013, Vol. 13 Issue 6, p935
In an earlier paper of ÄŒadek, VokÅ™Ãnek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the...