A topological space is a space defined independent of stretching or warping, up to tearing or gluing. A short line segment is the same as a long line segment. A line segment is distinct from a closed loop from a sphere from a cylinder.
A line segment is easy enough to recognize but a more exotic space less so. Instead the space can be characterized by dividing up into triangles and studying their properties, in particular their boundaries. Called singular homology classes.
N-dimensional triangles (called simplices) and boundaries are intuitive. A 0-dim. triangle is a point, 1-dim a line segment, 2-dim a right triangle, 3-dim. a tetrahedron. A boundary for a space is the corresponding lower-dim. space: for a line segment the difference between endpoints, for a 2-d circle its 1-dim. perimeter, etc. Some spaces have no boundary, like a closed loop.
Simplices
A simplex is an n-dim. triangle
\[\Delta^k = \{ (q_1, ...., q_k) \in \mathbb{R}^k \mid \sum_{i=1}^k q_i \leq 1; 0 \leq q_i \}\]\(\Delta^0\) is a dot, \(\Delta^1\) a line segment, \(\Delta^2\) a right triangle.
(crude ascii art)
\(\Delta^0\) *
\(\Delta^1\) ---
\(\Delta^2\)
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To work with them in a manifold \(M\) that is not \(\mathbb{R}^k\) define a corresponding smooth map
\[\sigma: \Delta^k \to M\]called the singular k-simplex.
k-chains are linear combinations of singular k-simplices, with integer coefficients \(a_i\)
\[c = \sum_i a_i \sigma_i\]defining a vector space, more generally an abelian group.
Projections and Boundaries
A projection onto a triangle gives one of its faces, itself a lower-dimensional triangle
\[F_i^{k}: \Delta^k \to \Delta^{k+1}\] \[F_i^{k}(q_1, ...q_k) := (q_1,..., q_{i-1}, 0, q_i, ...., q_k)\]For example a 2-dim simplex (right triangle) has three faces.
The corresponding projection into \(M\) is \(\beta^i = \sigma \circ F_i^{k}\), and the boundary operator \(\partial\) is constructed from projections
\[\partial \sigma := \sum_{i=0}^{k}(-1)^i \beta^i\]Defined on chains it transforms a k-chain into a k-1 chain, and is naturally a linear operator. From the definition the boundary of a line segment is the difference of its endpoints, the boundary of a right triangle is its (going counter-clockwise) perimeter path, etc. The \((-1)^i\) flips the path direction of corresponding faces so there is a consistent traversal in a single direction.
The boundary operator applied twice is zero
\[\partial \circ \partial = 0\]‘Proof’ by inspection: given a right triangle (singular 2-simplex) \(c\) the boundary operator applied once performs a counter-clockwise traversal of its perimeter, mapping a point in \([0, 1]\) to the sum of three vectors in \(M\) one for each face traversal. Each incremental traversal moves the triangle left triangle side down an increment, the bottom triangle side rightward an increment, and the diagonal triangle side increment up and to the left an increment. The sum of these vector increments is exactly zero. So \(\partial c\) is a singular simplex that maps to a single fixed point in \(M\). The boundary of a fixed point \(p\) is \(p - p\), or 0.
Singular Homology Classes
A k-chain with zero boundary is called a k-cycle, and a k-chain that can be expressed as the boundary of another (k+1)-chain is called exact.
Since all exact chains are cycles (\(\partial^2 c = 0\)), the respective vector spaces define a natural quotient space.
\[Z_k(M) := \{c \mid \partial c = 0 \}\] \[B_k(M) := \{c \mid c = \partial w \} \subset Z_k(M)\] \[H_k(M) := \frac{ Z_k(M) } { B_k(M) }\]\(H_k(M)\) is called the kth singular homology class. Homology classes characterize the corresponding topological space: homeomorphic topological spaces have isomorphic singular homology classes.