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A list of representatives of homology classes.

A complex is a vector of triples. Each index represents a generator, $x$, and the triple refers $(dx, d^{-1}x, alive?)$, where $dx$ is the list of generators that $x$ maps to, $d^{-1}x$ is the list of generators $y$ such that $x in dy$, and alive? indicates whether x is still an element of the complex (alive? = true by default).

>>> example1
IV.fromList [([], [3], True), ([], [3], True), ([], [], True), ([0, 1, 2], [], True), ([], [3], True)]

Compute the homology of a complex by the following method: Represent the complex by a directed graph, where the vertex set is the set of generators, and there is an edge from x to y if $y in dx$. For each $x$ such that $dx neq emptyset$, pick $y in dx$. For each $z in d^{-1}y$, and $w in dx$, if there is an edge from $z$ to $w$, delete it, otherwise create an edge from $z$ to $w$. Finally, delete $x$, $y$, and all edges into and out of $x$ and $y$. Continue iterating this process until there are no edges left, then read off the homology (the list of elements that are still alive in the complex).

>>> homology example1
[1,2,4]
>>> homology example2
[2,3,4,5]

An example complex.

>>> example1
IV.fromList [([], [3], True), ([], [3], True), ([], [], True), ([0, 1, 2], [], True), ([], [3], True)]

An example complex.

>>> example2
IV.fromList [([1,4],[],True), ([],[5,2,0],True), ([4,1],[],True), ([],[],True), ([],[5,2,0],True), ([1,4],[],True)]