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, and take the nerve (Delaunay complex).
is alternating sum over all simplices 
in Delaunay complex.
partition into intervals (for acyclic case only)
components, 
loops, 
voids.
is born when it is not in image of previous homology group.
[Cohen-Steiner, E., Hares 2007]. One of the most important theorems in persistent homology.
.
. [E., Nikitenko, Reitones, 2016]
template for a thesis. Quite nice and simple, and easily customizable.
and 
. The diagram then has the following form:
is replaced by a subgroup 
. The differentials 
go two units to the right, and we replace the term 
by the term 
, where the 
‘s refer to the 
now go diagonally upward because of the formula ![j_2(i_1a)=[j_1a]](https://s0.wp.com/latex.php?latex=j_2%28i_1a%29%3D%5Bj_1a%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. The maps 
and 
still go vertically and horizontally, 
and 
.
and a subspace 
, define 
. Since the boundary map 
takes 
to 
, it induces a quotient boundary map 
.
where 
holds. The relative homology groups 
are the homology groups 
of this chain complex.
– chains 
such that 
.
for some 
and 
.
is as follows: If a class ![[\alpha]\in H_n(X,A)](https://s0.wp.com/latex.php?latex=%5B%5Calpha%5D%5Cin+H_n%28X%2CA%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is represented by a relative cycle ![\partial[\alpha]](https://s0.wp.com/latex.php?latex=%5Cpartial%5B%5Calpha%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is the class of the cycle 
in 
.
is a nonempty closed subspace that is a deformation retract of some neighborhood in 
is the inclusion 
and 
is the quotient map 
.
and 
for 
.
take 
so that 
. The terms 
in the long exact sequence are zero since 
is contractible.
are isomorphisms for 
and that 
. Starting with 
, 
for 
, the result follows by induction on 
to be the homology groups of the augmented chain complex 
where 
. We require 
and 
, 
vanishes on 
and hence induces a map 
with 
. So 
. Clearly, 
for 
such that 
, the exact MV sequence has the form
with 
the northern and southern hemispheres, so that 
. Then in the reduced Mayer-Vietoris sequence the terms 
are zero. So from the reduced Mayer-Vietoris sequence 
we get the exact sequence 
.
}
as the group of all rotations about the origin of 
as the unit 3-sphere 
with antipodal points identified.
in 
to the rotation about 
(and maps 
to the identity rotation). This mapping is clearly smooth and surjective. Its restriction to the interior of 
. On the boundary of 
and through 
are the same. Hence the mapping induces a smooth bijective map from 
, with antipodal points on the boundary identified, to 
is also smooth. (To see that the inverse is smooth, write 
. Then 
, and so 
exists and is continuous for all orders 
. Similar results hold for the variables 
and 
, and also mixed partials. By multivariable chain rule, one can see that all component functions are indeed smooth, so the inverse is smooth as claimed.)
, the unit ball 
is a diffeomorphism between 
(diffeomorphic)
, we may view 
is well-defined since if 
, then 
.
, it is clear that 
. So 
, where 
denotes the set of 2 by 2 complex matrices.
, it is clear that 
and 
are smooth maps since their component functions are of class 
. Since 
and 
) are also smooth.
is monotonically decreasing from the top row down. For each fixed column 
is a constant. We have, 
. Hence, the degree of the elements in each column is monotonically increasing with row. That is, for fixed 
is monotonically increasing as 
, although the algorithm works for any field.![\mathbb{Z}_2[t]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D_2%5Bt%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
-module by the correspondence in previous Theorem. In this section we use 
and 
to denote homogeneous bases for 
and 
respectively.
since we are computing over 
is![\displaystyle M_1=\begin{bmatrix}[c|ccccc] &ab &bc &cd &ad &ac\\ \hline d & 0 & 0 & t & t & 0\\ c & 0 & 1 & t & 0 & t^2\\ b & t & t & 0 & 0 & 0\\ a &t &0 &0 &t^2 &t^3 \end{bmatrix}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+M_1%3D%5Cbegin%7Bbmatrix%7D%5Bc%7Cccccc%5D++%26ab+%26bc+%26cd+%26ad+%26ac%5C%5C+%5Chline++d+%26+0+%26+0+%26+t+%26+t+%26+0%5C%5C++c+%26+0+%26+1+%26+t+%26+0+%26+t%5E2%5C%5C++b+%26+t+%26+t+%26+0+%26+0+%26+0%5C%5C++a+%26t+%260+%260+%26t%5E2+%26t%5E3++%5Cend%7Bbmatrix%7D.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
of 
has the following basic property: 
provided 
.
relative to the standard basis for 
. We then reduce the matrix according to the reduction algorithm described previously.
, 
hence the standard basis may be used to represent 
.
and represent 
relative to 
and the homogeneous basis for 
by Gaussian elimination on the columns, using elementary column operations. From linear algebra, we know that 
is the number of pivots in the echelon form. The basis elements corresponding to non-pivot columns form the desired basis for 
on a manifold 
is said to be closed if 
, and exact if 
for some 
of degree one less.
, every exact form is closed.
be the vector space of all closed 
be the vector space of all exact 
.
in 
iff 
for some exact form 
.![[\omega]](https://s0.wp.com/latex.php?latex=%5B%5Comega%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
. Let 
be the free abelian group with basis the set of singular 
for 
and 
. A boundary map 
is defined by ![\displaystyle \partial_n(\sigma)=\sum_i(-1)^i\sigma|[v_0,\dots,\widehat{v_i},\dots,v_n].](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cpartial_n%28%5Csigma%29%3D%5Csum_i%28-1%29%5Ei%5Csigma%7C%5Bv_0%2C%5Cdots%2C%5Cwidehat%7Bv_i%7D%2C%5Cdots%2Cv_n%5D.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be a map in 
. We construct the mapping cone 
, where ![[1,x]\in CX](https://s0.wp.com/latex.php?latex=%5B1%2Cx%5D%5Cin+CX&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is identified with 
for all 
.
we have 
if and only if 
has an extension 
to 
.
has an extension 
.
) If 
by ![\tilde{h}(y_0,[t,x])=\psi[t,x]](https://s0.wp.com/latex.php?latex=%5Ctilde%7Bh%7D%28y_0%2C%5Bt%2Cx%5D%29%3D%5Cpsi%5Bt%2Cx%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, ![\tilde{h}(y,[0,x])=g(y)](https://s0.wp.com/latex.php?latex=%5Ctilde%7Bh%7D%28y%2C%5B0%2Cx%5D%29%3Dg%28y%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Note that ![\tilde{h}(y_0,[0,x])=\psi[0,x]=g(y_0)=z_0](https://s0.wp.com/latex.php?latex=%5Ctilde%7Bh%7D%28y_0%2C%5B0%2Cx%5D%29%3D%5Cpsi%5B0%2Cx%5D%3Dg%28y_0%29%3Dz_0&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Since ![\displaystyle \tilde{h}(y_0,[1,x])=\psi[1,x]=gf(x)=\tilde{h}(f(x),[0,x]),](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Ctilde%7Bh%7D%28y_0%2C%5B1%2Cx%5D%29%3D%5Cpsi%5B1%2Cx%5D%3Dgf%28x%29%3D%5Ctilde%7Bh%7D%28f%28x%29%2C%5B0%2Cx%5D%29%2C&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
induces a map 
which satisfies ![h[y,[0,x]]=\tilde{h}(y,[0,x])=g(y)](https://s0.wp.com/latex.php?latex=h%5By%2C%5B0%2Cx%5D%5D%3D%5Ctilde%7Bh%7D%28y%2C%5B0%2Cx%5D%29%3Dg%28y%29&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. That is 
.
) If 
, then define 
by ![\psi([t,x])=h[y_0,[t,x]]](https://s0.wp.com/latex.php?latex=%5Cpsi%28%5Bt%2Cx%5D%29%3Dh%5By_0%2C%5Bt%2Cx%5D%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. We have ![\psi([0,x])=h[y_0,[0,x]]=z_0](https://s0.wp.com/latex.php?latex=%5Cpsi%28%5B0%2Cx%5D%29%3Dh%5By_0%2C%5B0%2Cx%5D%5D%3Dz_0&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Then ![\displaystyle \psi([1,x])=h[y_0,[1,x]]=h[f(x),[0,x]]=gf(x).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cpsi%28%5B1%2Cx%5D%29%3Dh%5By_0%2C%5B1%2Cx%5D%5D%3Dh%5Bf%28x%29%2C%5B0%2Cx%5D%5D%3Dgf%28x%29.&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
That is, 
.
is a smooth manifold.![U_1=\{[z^1, z^2]\mid z^1\neq 0\}](https://s0.wp.com/latex.php?latex=U_1%3D%5C%7B%5Bz%5E1%2C+z%5E2%5D%5Cmid+z%5E1%5Cneq+0%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![U_2=\{[z^1, z^2]\mid z^2\neq 0\}](https://s0.wp.com/latex.php?latex=U_2%3D%5C%7B%5Bz%5E1%2C+z%5E2%5D%5Cmid+z%5E2%5Cneq+0%5C%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
. Also define 
by ![g_1([z^1, z^2])=\frac{z^2}{z^1}](https://s0.wp.com/latex.php?latex=g_1%28%5Bz%5E1%2C+z%5E2%5D%29%3D%5Cfrac%7Bz%5E2%7D%7Bz%5E1%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![g_2([z^1, z^2])=\frac{\overline{z^1}}{\overline{z^2}}](https://s0.wp.com/latex.php?latex=g_2%28%5Bz%5E1%2C+z%5E2%5D%29%3D%5Cfrac%7B%5Coverline%7Bz%5E1%7D%7D%7B%5Coverline%7Bz%5E2%7D%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be the homeomorphism from 
to 
defined by 
and define 
by 
.
is an open cover of 
are well-defined homeomorphisms (from 
onto an open set in 
is an atlas of 
![\begin{aligned} \phi_2\phi_1^{-1}(x,y)&=\phi_2 g_1^{-1}(x+iy)\\ &=\phi_2([1,x+iy])\\ &=f(\frac{1}{x-iy})\\ &=f(\frac{x+iy}{x^2+y^2})\\ &=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}) \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++%5Cphi_2%5Cphi_1%5E%7B-1%7D%28x%2Cy%29%26%3D%5Cphi_2+g_1%5E%7B-1%7D%28x%2Biy%29%5C%5C++%26%3D%5Cphi_2%28%5B1%2Cx%2Biy%5D%29%5C%5C++%26%3Df%28%5Cfrac%7B1%7D%7Bx-iy%7D%29%5C%5C++%26%3Df%28%5Cfrac%7Bx%2Biy%7D%7Bx%5E2%2By%5E2%7D%29%5C%5C++%26%3D%28%5Cfrac%7Bx%7D%7Bx%5E2%2By%5E2%7D%2C%5Cfrac%7By%7D%7Bx%5E2%2By%5E2%7D%29++%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is of class 
is a smooth manifold.
and 
. Then 
is an open cover of 
by 
and 
by 
.
. Hence 
is a homeomorphism from 
onto an open set in 
is a homeomorphism from 
onto an open set in 
is an atlas for 
is differentiable of class 
are of class 
is of class 
.
by 
and 
.
is well-defined since if 
then 
so that 
.
defined by 
and 
.
is smooth. Hence 
. A linear map 
is called a derivation at 
if it satisfies 
, is defined as the set of all derivations of 
at 
