Delta, Nerve, Čech complexes
01 Theory - Delta complexes
The
This technique allows many common spaces to be represented using fewer simplices. For example, a circle may be given as a
Why not simplicial?
These spaces cannot be formed as simplicial complexes by gluing the given simplices.
They are not literally simplicial complexes because the circle is round and the torus is not a subset of
. They are not homeomorphic to simplicial complexes (with the indicated decompositions) because the circle has a 1-loop while the torus has simplices which intersect in more than one face. (Both are impossible features for the homeomorphism type of simplicial complexes with the given decompositions.)
-Complex - Combinatorial A combinatorial
-complex is a sequence of simplex sets together with attaching maps , for , one for each facet of a simplex in . Attaching maps join the faces of each simplex to simplices of lower dimension. This works geometrically when the attaching maps preserve adjacency of faces of faces. Preserving adjacency is equivalent to this combinatorial condition:
The adjacency condition may be illustrated with a 2-simplex:
Faces are numbered by removing a vertex in the order of vertices. So
Now interpret the adjacency condition: Take as input the 2-simplex.
- With
- The condition says “vertex 0 of edge 1 equals vertex
of edge 0.” - These vertices are the point
.
- The condition says “vertex 0 of edge 1 equals vertex
- With
- The condition says “vertex 0 of edge 2 equals vertex
of edge 0.” - These vertices are the point
.
- The condition says “vertex 0 of edge 2 equals vertex
- With
- The condition says “vertex 1 of edge 2 equals vertex
of edge 1.” - These vertices are the point
.
- The condition says “vertex 1 of edge 2 equals vertex
-Complex - Topological A
-complex is the topological space formed from a combinatorial -complex as the disjoint union of simplices enumerated by elements of , glued together according to the combinatorial attaching maps with canonically defined linear functions interpolating between glued vertices. where
, with is the inclusion of the face of the standard simplex , and is any point in that face.
Simplicial vs.
complexes How best to think about simplicial vs.
- complexes?
- Simplicial complex:
- Collection of geometric simplices in the same background space
. - A sub-simplex brought to touch another sub-simplex only in one congruent face, where they are glued.
-complex:
- Collection of abstract geometric simplices in distinct background spaces (disjoint union).
- Simplices in
are attached to the lower-dimensional structure by gluing in the faces of new -simplices, preserving only the adjacency relation.
Underlying space
Given a complex
(simplicial complex or -complex), the underlying topological space of is denoted .
-homology Homology may be defined for
-complexes using the same technique as for simplicial complexes. When a simplicial complex is also a
-complex, their homology groups are canonically isomorphic.
02 Theory - Nerve complex
Nerve complex
Suppose
is a collection of sets indexed by . The nerve complex is the collection of finite subsets such that is nonempty. That is, the nerve is the complex of indices of finite collections of mutually overlapping subsets from
.
Notes:
- The definition does not assume that the
are subsets of a topological space. If they are, we do not assume that they are open. - The nerve complex is a combinatorial complex: the elements are sets of indices, not the subsets themselves.
- Suppose
is a subset in the nerve. - Then all subsets of
are also in the nerve. - I.e. the nerve is closed under taking subsets.
- Then all subsets of
- Therefore: the nerve is a combinatorial simplicial complex.
As a simplicial complex:
- vertices of the nerve correspond to individual sets
-simplices in the nerve correspond to nonempty subsets having elements
Theorem: Geometric realization
Every combinatorial simplicial complex with
vertices can be realized as a geometric simplicial complex in . In fact, every combinatorial simplicial complex with maximum simplex dimension equal to
can be realized as a geometric simplicial complex in .
03 Theory - Čech complex
Čech complex
Suppose
is a metric space. Suppose is a finite set of points in . Fix
. Define as the open balls: The nerve of
is called the Čech complex of .
04 Illustration
Let
The Čech complex created using
The Čech complex created using
Homotopy type
05 Theory - Homotopy type and homology
Homology can be used to distinguish topological spaces or to measure the topological complexity of a space. From the perspective of computational complexity theory, homology is very easy to compute.
Theorem: Homology depends functionally on
The isomorphism types of the homology groups
of a simplicial complex depend only on , the underlying topological space. More precisely, a homeomorphism of topological spaces induces an isomorphism of homology groups.
This theorem implies that any two homeomorphic spaces have isomorphic homology groups.
- %& For a topological space
, we write for the homology groups of computed using any simplicial complex structure provided on . - If
does not have a simplicial complex structure, or -complex structure, one can still define singular homology for , and this homology still depends functionally only on the homeomorphism type. - Singular homology is isomorphic to simplicial and
-homology when the latter can be defined.
- If
There is an equivalence relation between spaces that is much ‘coarser’ than homeomorphism, namely homotopy equivalence.
Homotopy equivalence
A homotopy equivalence between topological spaces
and is a pair of continuous maps:
such that
is homotopic to and is homotopic to . We write
when and are homotopy equivalent.
A homotopy from
A homeomorphism is actually the special case of a homotopy equivalence where the homotopies
Theorem: Homology depends functionally on homotopy type
The isomorphism types of the homology groups
of a topological space depend only on the homotopy type of the space. More precisely, a homotopy equivalence of topological spaces induces an isomorphism of homology groups.
We will see a partial justification of this fact later, after discussing simplicial maps and chain maps.
Contractible spaces
A topological space is contractible if it is homotopy equivalent to a point.
06 Illustration
Contractibility of
We show that
is contractible. Let
be the origin. Consider the maps and :
Then
is the constant map sending everything in to the origin. Define
. Then , which agrees with , while . Therefore is homotopic to . Now consider that
already. No additional homotopy is needed.
The same homotopy formula, namely
- ! Therefore, any simplex is contractible.
Cylinder is homotopic to circle
A cylinder is given by
. Consider the ‘collapse’ map given by projecting to the circle along straight lines, and the map given by inclusion:
The map
is equal to the collapse map, and the map is the identity on . Now construct a homotopy from the collapse map to
:
Rank of homology
07 Theory - Betti numbers, Euler Characteristic
Rank of an abelian group
Every finitely generated abelian group
is isomorphic to a group with this form:
- The
summands give a free abelian part and contain all infinite-order generators. - The
summands give torsion and contain all finite-order generators. The rank of
is the number .
Betti numbers
The Betti numbers
of a topological space are the ranks of the homology groups:
These Betti numbers may be combined into a single number, the Euler characteristic:
Euler characteristic
The Euler characteristic
of a topological space is the alternating sum of Betti numbers:
08 Illustration
For convex polyhedra, Euler’s formula for the Euler characteristic is simply vertices minus edges plus faces:
For nonconvex polyhedra, this formula can give values other than the actual Euler characteristic defined using Betti numbers.
| Name | Figure | Vertices | Edges | Faces | |
|---|---|---|---|---|---|
| Tetrahedron |  | 4 | 6 | 4 | 2 |
| Cube |  | 8 | 12 | 6 | 2 |
| Octahedron |  | 6 | 12 | 8 | 2 |
| Dodecahedron |  | 20 | 30 | 12 | 2 |
| Icosahedron |  | 12 | 30 | 20 | 2 |
| Tetrahemihexahedron |  | 6 | 12 | 7 | 1 |
| Cubohemioctahedron |  | 12 | 24 | 10 | -2 |
| Small stellated dodecahedron |  | 12 | 30 | 12 | -6 |
For a simplicial complex, the Euler characteristic can always be computed using a formula analogous to the Euler formula: count the simplices of each dimension:
Here
Theorem: Euler
Euler For a simplicial complex, the same Euler characteristic may be calculated using Betti numbers or using the counts of simplices.
We do not prove this here, although the proof is a straightforward homology calculation.
Maps and homology
09 Theory - Simplicial maps, chain maps
Simplicial map
A map
between simplicial complexes is simplicial if it maps vertices of to vertices of , and it maps simplices of linearly to simplices of .
For example:
- Inclusions of one simplicial complex into another.
- Reduction of a simplicial complex to lower dimension, e.g.
.
Chain map
A chain map
is a map between chain complexes, which consists of a collection of homomorphisms: of abelian groups which commute with the boundary operators:
Such a chain map makes the following diagram commute:
- The homomorphisms can be collected under a single map
, and the commutativity condition expressed simply:
Theorem: Induced chain map
A simplicial map
determines a map of chain complexes . The simplicial map
is extended linearly from simplices to chains (from generators to linear combinations).
10 Theory - Induced maps on homology
Theorem: Induced map on homology
A chain map
determines a map of homology groups which splits by degrees: A chain map
sends cycles to cycles and boundaries to boundaries, so it descends to a map of quotient groups.
Proof of Theorem: Induced map on homology
It is sufficient to check that
maps cycles to cycles and boundaries to boundaries:
- By mapping boundaries to boundaries,
is well-defined on homology groups. - By mapping cycles to cycles,
maps elements of homology to elements of homology.
Checking
maps cycles to cycles: If
, meaning that is a cycle, then for some , whence: but this implies
.
Checking
maps boundaries to boundaries: If
, then: but this means the image of
is a cycle.
Proposition: Homotopic maps are the same on homology
If two maps of simplicial complexes
are homotopic, then they induce the same maps on homology groups, :
In words: continuous deformations do not change the map induced on homology.
Proposition: Homology respects composition of maps
Given a composition of simplicial maps:
the map induced by the composite agrees with the composite of the induced maps:
The property of respecting composition of maps is called functoriality.
Corollary: Homotopy equivalence implies homology isomorphism
Homotopy equivalent simplicial complexes have isomorphic homology groups:
Proof of Corollary
Suppose we have a homotopy equivalence:
Now apply functoriality to the induced maps on homology:
Therefore
is the inverse of as homomorphisms of abelian groups.
11 Illustration
Contraction to a point
The map
for some point is a homotopy equivalence, so it induces an isomorphism on homology. Therefore:
Dunce Hat
The Dunce Hat is formed by identifying all sides of a triangle with non-circular identifications:
The Dunce Hat is contractible, but it is not collapsible. The contraction can be performed by embedding the Hat in a 3-ball in
. Because it is contractible, the Dunce Hat has the homology of a point.
Homology of a cylinder
Recall that the cylinder is homotopic to the circle (§06):
A circle is homeomorphic to a triangle.
We have computed the homology of a triangle (§11 of Part I).
Therefore the homology of the cylinder is:
Homology variations
12 Theory - Relative homology
Suppose a simplicial complex
More precisely, suppose that
Then observe that chains in
(A subgroup: because a sum of chains in
Relative chain complex
Suppose
is an inclusion of simplicial complexes. Define
, the relative chain group, as the quotient:
Proposition: Boundary maps descend to relative chain groups
The boundary maps
send chains in to chains in and thus descend to the quotient groups, giving a relative chain complex: (Here
is just the map induced by on the quotient groups.)
Proof of Proposition: Boundary maps descend to relative chain groups
It is sufficient to establish that boundaries from
must also reside in , i.e. that: But this is clear because
is a simplicial complex, so it is closed under taking boundaries.
Relative homology
Suppose
is an inclusion of simplicial complexes. Define
, the relative homology group, as the homology group of the relative chain complex:
Remark: Relativizing before homology
It is tempting to try relativizing the homology groups directly.
The inclusion
does induce maps . We could study the groups . These quotient groups are not what relative homology is after! See the illustrations.
13 Illustration
Homology of a simplex relative to its boundary
Consider a simple 2-simplex
and the simplicial complex that it determines:
Let
be the boundary 1-simplex (the edges only from ). There are no 2-simplices in
, so: All 0- and 1-simplices in
are also in , so: The relative chain complex:
becomes:
It follows that the relative homology is:
Compare this to the homology of a sphere!
Homology of a sphere:
Notice that
is homeomorphic to the quotient space . This comparison of homologies is apt; we just need to explain
versus .
Compare this against the quotient of homology groups!
The homology of
is: The homology of
, because it is contractible, is: The quotient of homology groups is:
- Relative homology treats
as a single point. - The quotient of homology annihilates generators (“holes”) in
which are also generators (“holes”) in . - Different results for different constructions!
14 Theory - Cochain complex
Recall that the chain group
One should think of the chains
Thinking in this way, the boundary map
Cohomology is created by interpreting the integers
To notate this effectively, define indicator functions
Cochains
The space of linear maps
is isomorphic to the free abelian group of formal linear combinations of indicator functions: Elements of this space, written generically as formal sums
, are called cochains. (For cochains it is not necessary that
be finite.)
Since there is a bijective correspondence of sets between oriented simplices and their indicator functions, we see that
In both cases, the data of a chain / cochain consists of the assignment of a number (
Coboundary operator
Define the coboundary operator thus:
In words: the coboundary of a simplex indicator indicates that simplex in the boundary of a higher simplex.
For example, in the triangular 2-simplex:
If
Thus, in general,
Witness a simple computation, applying
Cochain complex
Given a simplicial (or
-) complex of maximal dimension , we define the cochain complex of as the sequence of morphisms: This sequence is a complex because:
The image and kernel groups are named in the obvious way:
— called the coboundary cochains — called the cocycles
15 Theory - Cohomology
Cohomology
The
cohomology of is defined as the quotient group of cocycles modulo coboundary chains:
Cohomology is an upgrade from homology because it has an additional product structure, so cohomology actually gives a collection of rings.
Cup product on cochains, for simplicial complexes
Suppose we are given a simplicial complex
, together with a chosen ordering of all vertices of . There are maps of cochain groups
defined as follows: For any pair of cochains
and , and any vertices ordered according to , with the -simplex they form, we have: Note: the oriented simplices here may be written with various orderings, but the splitting into
- and -simplices must follow the order of .
Cup product using indicators
Let
be the indicator of an oriented simplex , and let be the indicator of another oriented simplex . Let
be the vertices of the total list reordered in the ordering of
(allowing repetition), so . Then
is defined as follows: it is the indicator of provided:
- (i) the
are all distinct, - (ii)
for all ; and
if (i) and (ii) do not hold. The definition is extended bilinearly to all of
.
Vertex orderings and the cup product
Instead of choosing a total ordering
of all vertices of , it is sufficient to choose a partial ordering of the vertices of having the property that induces a total ordering of each simplex of . A set of compatible orderings of the vertices in each simplex is used to define a ‘front half’ and ‘back half’ of each
-simplex , thereby cutting into one -simplex and one -simplex. The cutting method applied to a given simplex must be compatible with the method applied to every other simplex. A global vertex ordering ensures this compatibility.
Cup product on cohomology
The cup product has the following relation to the coboundary operator, a type of signed Leibniz rule:
when
. This formula implies that if
and are cocycles (or coboundaries), then is also a cocyle (coboundary). Therefore the cup product descends to cohomology: The cup product on cohomology provides a ring structure to the direct sum
. The product, and the ring structure, do not depend on the choice of ordering used to define the product on cochains. (Note: The cup product is always zero when
is greater than the maximal simplex dimension.)
16 Illustration
Some cohomology classes on the 2-torus
Consider the 2-torus
:
The red (poloidal) and pink (toroidal) curves together generate the degree one homology,
. Now recall a simplicial complex representation of the torus:
Let
be the cochain which assigns 1 to every edge in the middle column, and zero to all other edges. (So it is 1 on the edges directly below the label, including the diagonals, but excluding all verticals.) Let be the cochain which assigns 1 to every edge in the middle row, and zero to all other edges. (So it is 1 on the edges directly to the right of the label, including the diagonals, but excluding all horizontals.) Accounting now for orientations, let give on edges directed rightwards, and give on edges directed upwards.
One can check directly, going around the boundaries of each 2-simplex, that
and : The sum of the numbers on any triangle, accounting for orientations, is zero. So these cochains are both cocycles. In order to see that
and are not boundaries, and thus represent nontrivial homology classes, and also to see that they don’t represent the same homology class, we devise a method of representing coboundaries of 0-cochains. A 0-cochain
is the assignment of integers to each of the vertices. The condition , which means , states that these numbers cannot change as one moves across an edge from vertex to vertex. (Applying to any edge gives . By linearity, . Therefore when the condition holds.) From this it follows that
. The isomorphism is given by sending to the integer it gives on a vertex. Return now to
. A boundary in is a cochain assigning numbers to directed edges such that there is a vertex labeling cochain with . In other words, that it is possible to derive the edge numbers as the differences of a single vertex numbering scheme. Now one can see that
is not a coboundary. Vertex integers must increase by one as one passes across the middle column of edges, and stay fixed going across all other edges. This is impossible because the left-most and right-most edges are identified. Similarly is not a coboundary, and is not a coboundary. (To witness a contrasting case, let
be a 1-cochain giving 1 only on the non-vertical edges in the right-most column. This is shifted to the right. Now is a coboundary: assign 1 on vertices in the middle-right column, and 0 on all other vertices.) Much more work would be required to show that
and have no relations in homology, and that they jointly generate homology.
A cup product on the 2-torus
We study the cup product
:
We must provide an ordering
of all the vertices of the torus. Order them as in a book: top-to-bottom, then left-to-right.
Let us see how to verify the values on the green and yellow 2-simplices. Label the vertices of the inner square as in the figure; the index ordering is compatible with
. For the green 2-simplex
, we have and , so: For the yellow 2-simplex
, we have and , so: A calculation with more details (illustrating bilinearity) can be performed in terms of the indicator functions. Write:
where
is the indicator for the edge in the support of which is in the place from the top (oriented rightwards), and analogously is for the one for in the place from the left (oriented upwards). Applying bilinearity to expand the cup product:
The edges with indicators
and could span one of the 2-simplices only when and . And in fact the pairs can be ruled out as well. The only terms to consider are these six:
- Now
is the indicator for and is the indicator for , so because so the total list does not split according to the ordering of . - Then
is the indicator for , so is the indicator for , meaning it returns on the green 2-simplex. - Then
is the indicator for , so . - Then
is the indicator for , so because . - Then
is the indicator for , so because . - Then
as well, also because . By adding these results, we see that
is the indicator for the central 2-simplex, oriented clockwise. This cochain in
is automatically a cocycle because is empty. Is it a coboundary?
To see that the answer is “no,” we can devise another scheme to represent coboundaries, now for
-simplices. We give here only a sketch. Give a 1-cochain which assigns integers to the edges. Draw oriented paths crossing the edges, with the number of paths crossing matching . Then for a triangle counts the total flow of paths into or out from ; it is a ‘sink’ or ‘source’ number describing extra paths originating or terminating inside . The condition would mean that the paths are conserved in , none created or destroyed. With this perspective, it is obvious that the cochain
depicted in the figure cannot be a coboundary. Every 2-cochain which is a coboundary describes a collection of oriented paths that are created or destroyed only in triangles with nonzero assignments, according to that assignment. So would describe a single path terminating in the green triangle, but originating nowhere, since there is no other sink or source. Another observation is worth making about
. It is the generator of . Note that the full cohomology of
is given by: Here is a fact: All indicator functions on 2-simplices determine the same cohomology class. To see this, take the difference of any two indicators. Draw a simple oriented path out of one of them into the other. This path will define a 1-cochain by assigning
on any edge it crosses on the way (according to their mutual orientations), and 0 on all edges it does not cross. The coboundary of this cochain gives on the triangle from which the path emanates, and on the triangle in which the path terminates, and zero on all other triangles. That is exactly the difference of indicator 2-simplices we started with. Since
is generated by indicator 2-simplices, and they are all equivalent, we deduce that it is generated by . To see that it is in fact infinite and isomorphic to
, we must exclude relations among the multiples of . A relation is equivalent to the relation for . But we can exclude any such relation by the reasoning used above: a cochain cannot be a coboundary because it would represent oriented paths terminating in the green 2-simplex, and originating nowhere. Therefore we have proven that:
The cocycle
is called the fundamental cocycle of .
Cohomology ring has topological significance
Let
, the 2-torus. Let
, a 2-sphere with two circles attached at one common point. The cohomology groups are isomorphic:
However, the cup product operations differ, so the ring structures differ.
- In
, the cup product of the two generating cocycles is the fundamental coclass of the 2-torus. - In
, the cup product of the two generating cocycles (dual to the two copies of ), is zero.
17 Theory - Coefficients
It is possible to define homology and cohomology with other coefficients. Given an abelian group
And the cochain groups thus:
Homology and cohomology are defined in the usual way from these chain groups.
The most important cases are
Manifolds
18 Theory - Topological and smooth manifolds
Topological manifold
A (second-countable, Hausdorff) topological space
is an -dimensional (topological) manifold if every point has an open neighborhood which is homeomorphic to .
Notes:
- An equivalent definition is that every
has open neighborhood homeomorphic to the open ball , since of course is itself homeomorphic to . - It is a theorem that every
-manifold can be embedded in some Euclidean space , so an equivalent definition states adds the hypothesis that for some .
To define smooth manifolds, we must first define differentiable and smooth maps.
Regular points
Suppose
with open. Let give coordinates on . Suppose
is a function given in coordinates by . The differential matrix of
at is the Jacobi matrix: The point
is regular for when has rank . (No point can be regular when .)
Smooth manifold in
A subspace
is a smooth -manifold if every has an open neighborhood in (open in the subspace topology) that is the image of a smooth injective function , whose domain is an open set , and whose differential has maximal rank everywhere on . Each such function is called a parametrization or a coordinate chart for
, and the image of a parametrization is a coordinate patch on .
A parametrized surface is a bijective and infinitely differentiable (smooth) injective map:
where the parameter domain
Using coordinates
The maximum rank of
- The columns of
are linearly independent (for two columns: not collinear vectors) - There are two linearly independent rows in
- There is a
submatrix of with nonzero determinant
If the matrix
The condition
19 Illustration
Wild topological manifolds
Topological spaces can be stranger than one imagines on the basis of simple examples. Here is Alexander’s Horned Sphere (/media/File:Construcción_de_esfera_de_Alexander_con_cuernos.gif; explanatory essay):
The horned sphere
is a subset of homeomorphic to . The outside contains non-collapsible loops (so it is not simply connected), which is unlike the standard inclusion of in . The map
with is an embedding of topological spaces and thus topological manifolds. (It is continuous, injective, and induces a homeomorphism onto the image which is endowed with the subspace topology; so the image is homeomorphic to as a topological manifold.) The map
is not an embedding of differentiable, smooth, or piecewise linear manifolds.
Parametrized paraboloid
Let
by .
Then
is given by: Notice that the determinant of the upper
square is nonzero.
Singular curve
Consider
given by :
Then:
and .
Parametrized 2-sphere
Spherical coordinates define a parametrization of the 2-sphere:
The differential:
Notice that poles of the sphere correspond to points with
, and there the right column is all zeros. So poles are not regular points of . One can check that every other point is regular by computing the cross product of the column vectors. Also, at the poles the function is not injective.
Therefore one must restrict the domain, for example to
and , to get a true parametrization. Unfortunately this does not cover the entire sphere; indeed it is impossible to cover the sphere with a single parametrization.
Parametrized 2-torus in
The 2-torus can be parametrized in
very easily:
This parametrization is regular everywhere, but the domain cannot be extended to cover the entire torus without becoming non-injective (and no longer a homeomorphism onto its image), for topological reasons. The torus cannot be covered by a single parametrization.
