This thesis is a collection of work under the theme of “applied topology."  The linking idea behind seemingly disjoint fields is the existence of a filtration that one uses to study a space. In turn, given the ubiquitous nature of filtrations, applications range from theoretical fields (e.g. symplectic geometry) to applied fields (machine learning).

In paper A, we study when homological information of a simplicial complex can be determined from its components in the following manner: given a data cloud, partition the points in the cloud into two (or more) sets. Form separate simplicial complexes from these sets, and compare the homologies of these simplicial complexes from that of the simplicial complex formed from the point cloud itself. In applied topology, very rarely does a decomposition of a space yield information about the space itself - meaning that it is rare for a Mayer-Vietoris sequence to hold. We study “obstruction complexes" and show that in nice enough cases, there is a relationship between homological information of the space and its decomposition.

In paper B, we study a construction called “realisation" that we apply to posets. This enables the generation of a wealth of examples of posets that might not necessarily be the nonnegative reals in topological data analysis. We define various properties of these realizations, and in the end we link these properties to homological properties of the functors that are being studied.

In paper C, we study the classic evasion-path problem. This problem is well-known in robotics and machine learning, and more recently became of interest in the applied topology community through works of Krishnan and Ghrist in addition to work of Adams and Carlsson. The key point is that just studying homology and barcodes could not determine if an evasion path exists. We study a higher invariant, using tools of Goodwillie calculus to yield an obstruction to the existence of an evasion path.

In symplectic geometry, work has been done to try to use the filtration to study symplectic embeddings. The work in this thesis does not get to the direct relationship between the filtration and symplectic embeddings, but it does study the relationship between symplectic embeddings of ellipsoids and polydiscs in dimension four, yielding a rigid-flexible result similar to the one given by the famous nonsqueezing theorem. This is the topic of paper D. There is still much work to be done linking applied topology and symplectic geometry.

Motivated by applications in Topological Data Analysis, we consider decompositionsof a simplicial complex induced by a cover of its vertices. We study how the homotopytype of such decompositions approximates the homotopy of the simplicial complexitself. The difference between the simplicial complex and such an approximationis quantitatively measured by means of the so called obstruction complexes. Ourgeneral machinery is then specialized to clique complexes, Vietoris-Rips complexesand Vietoris-Rips complexes of metric gluings.

We introduce a construction called realisation whichtransforms posets into posets. We show that realisations shareseveral key features with upper semilattices which are essentialin persistence. For example, we define local dimensions of pointsin a poset and show that these numbers for realisations behavein a similar way as they do for upper semilattices. Furthermore,similarly to upper semilattices, realisations have well behaved discrete approximations which are suitable for capturing homologicalproperties of functors indexed by them. These discretisations areconvenient and effective for describing tameness of functors. Homotopical and homological properties of tame functors, particularlythose indexed by realisations, are discussed.

We give an explicit description of the quadratic approximation of theidentity functor on the category of unpointed spaces. To do this, we construct a version of the stable James-Hopf map that is valid for unpointedspaces. While previous constructions of the James-Hopf map relied onconfiguration space models of Ω∞Σ∞X, our approach uses equivariantstable homotopy theory.By way of “application” we use the quadratic approximation to showan example of two fiberwise spaces that are stably fiber homotopy equivalent, but are not unstably fiber homotopy equivalent. The difference isdetected on the level of quadratic approximations. We use a simplifiedversion of an example proposed by Henry Adams in the context of thepath evasion problem.