In this tutorial lecture, we’ll introduce the foundational ideas of simplicial homology and Delta complexes, two powerful tools for understanding the shape and structure of spaces using algebra. We’ll learn how to build spaces out of simple building blocks like points, line segments, triangles, and higher-dimensional simplices, and how to systematically count “holes" of various dimensions using chains, boundaries, and cycles. Along the way, we’ll see how Delta complexes generalize simplicial complexes to allow more flexible constructions, and how homology groups capture essential topological features. Through hands-on examples and computations, students will gain intuition for how algebra can be used to detect the hidden features of topological objects.
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