A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology
Figure 7
Examples of low-dimensional manifolds and their Betti numbers with some of the corresponding loops.
(a) A point is a 0D loop; no higher dimensional loops are present. Thus, each manifold containing at least one point has a 0D loop, so every list of Betti numbers starts with a “1”. (b) A circle is a 1D loop, with no other loops in higher dimensions. (c) A 2D torus with two examples of non-contractible (red) 1D loops, and an example of a 1D loop contractible into a point (green). The 2D surface of the torus is the 2D loop listed. (d) A 2D sphere, with two exemplary contractible 1D loops. The 2D surface of the sphere “loops” onto itself.