User:Erel Segal/Matveev on Algebraic Topology

This is a short summary of the book:

Matveev, Sergey V. (2006). Lectures on Algebraic Topology. European Mathematical Society. ISBN 978-3037190234.

The summary does not contain the many figures and exercises in the book.

Elements of homology theory

categories and functors

Category with morphism is a generalization of many things: sets and maps, groups and homomorphisms, Abelian groups and homomorphisms, topological spaces and continuous maps, topological spaces and classes of homotopic maps. Each category also has its own isomorphisms.

There are covariant functors and contravariant functors between categories.

Homology theory is a functor from topology to algebra:

• from the category of topological spaces,
• to the category of sequences of Abelian groups.

To find whether two topological spaces X,Y are equivalent:

• Take a functor F from from the category of topological spaces to a category of groups.
• Compare the groups F(X),F(Y). If they are distinct, then X,Y are distinct (otherwise, nothing can be said).

For general topological spaces, calculating this functor may be difficult. For special cases, it may be easier. We focus on the special case of simplicial complexes.

Some geometric properties of $\mathbb {R} ^{n}$

The bases of $\mathbb {R} ^{n}$ are partitioned to two classes which are called orientations. There are exactly two orientations: + (right) and - (left).

Each n-dimensional simplex is contained in $\mathbb {R} ^{n}$, so we can assign to it an orientation by selecting a base for its containing plane.

The orientation of an n-dimensional simplex induces an orientation on each of its (n-1)-dimensional faces, using the rule of inward normal. Choose a base for the n-dimensional simplex representing its orientation, such that:

• Its first n-1 vectors lie on its (n-1)-dimensional face;
• Its n-th vector points into the simplex.
• Then, the first n-1 vectors determine the orientation of the face.

• • • See Simplicial homology#Definition for alternative definition of orientation based on permutations.

Theorem of doubly-induced orientation: Given an n-dimensional simplex S with a fixed orientation. It has two (n-1)-dimensional faces, S1 and S2. These faces are adjacent - they have a common (n-2)-dimensional face, S12. Calculate an orientation for S12 in two ways:

• Calculate the orientatation induced from S to S1, and then from S1 to S12.
• Calculate the orientatation induced from S to S2, and then from S2 to S12.

The results in the two ways are opposite!

Example for n=3: S is a tetrahedron, S1 and S2 are adjacent triangular faces, and S12 is their common segment. The orientation of S induces a cycle around S1 and a cycle around S2, and these cycles cross S12 in opposite directions.

It is important to distinguish between:

• A simplicial complex - K - a collection of simplices;
• Its underlying space - |K| - the union of the simplices. It is a polyhedron, and it is triangulated by K.

An orientation of a simplicial complex is a vector of orientations to each of its simplices (including their faces). For example, a triangle has 7 faces (itself, its three sides and its three vertices), so it has 128 distinct orientations.

To each simplicial complex, we can construct its homology groups in two steps:

• Assign to each simplicial complex, a chain complex (subsection 1.4)
• Assign to this chain complex, its homology groups (subsection 1.3)

Chain complexes

Example 1

.... ----> 0 --[d3]--> Z^2 --[d2]--> Z --[d1]--> 0 ----> ....

Dimension	Chains	Cycles	Boundaries	Homology group
3+	0	0	0	0
2	Z^2	(-1,1)*Z	0	Z
1	Z	Z	3 Z	Z_3
0-	0	0	0	0

Example 2 - Elementary complex E(m)

.... ----> 0 --[d_{m+1}]--> Z --[d_m]--> 0 ----> ....

Dimension	Chains	Cycles	Boundaries	Homology group
m+1+	0	0	0	0
m	Z	Z	0	Z
m-1-	0	0	0	0

Example 3 - Elementary complex D(m,k)

.... ----> 0 ----> Z --[d_{m+1}]--> Z --[d_m]--> 0 ----> ....

Dimension	Chains	Cycles	Boundaries	Homology group
m+2+	0	0	0	0
m+1	Z	0	0	Z
m	Z	Z	k*Z	Z_k
m-1-	0	0	0	0

Direct sum

The direct sum of two chain-complexes is calculated by taking the direct-sum of each chain-group.

The homology groups of a direct sum are the direct sum of the homology groups.

So we can start with elementary chain complexes and build more complicated ones using direct sums.

homology groups of a simplicial complex

We are given an N-dimensional simplicial complex, and it is oriented (i.e, for each simplex we select one of its two possible orientations). Define:

• C_n = the free Abelian group generated by the n-dimensional simplices of K (this group is non-trivial only for 0 <= n <= N).
• d_n = a homomorphism from C_n to C_{n-1} that takes each n-dimensional simplex to a sum of its faces, where the sign of each face depends on its orientation: the sign is + if the induced orientation of the face as a part of its parent equals the orientation of the face in K, and - if the orientations are reversed.

The theorem of doubly-induced orientation implies that the boundary of a boundary is empty, as required by the definition of chain complex.

The homology groups of a simplicial complex K do not depend on the orientation of K.

The homology groups of a polyhedron do not depend on the triangulation.

It is possible to define homology groups for topological polyhedra, that can be triangulated into curvilinear simplices.

Calculating homology groups of a topological space:

1. Present the space as a polyhedron and triangulate it.
2. Choose an orientation for the simplicial complex.
3. Calculate the chain groups C_n.
4. Describe the boundary homomorphism d_n.
5. Calculate the groups of cycles A_n.
6. Calculate the groups of boundaries B_n.
7. Calculate the quotient groups H_n = A_n/B_n.

Examples:

Point:

Dimension	Chains	Cycles	Boundaries	Homology group
0	Z	Z	0	Z

Segment:

Dimension	Chains	Homomorphism	Cycles	Boundaries	Homology group
1	Z	(1) -> (1,-1)	(0)	(0)	0
0	Z^2	0	Z^2	(1,-1)*Z	Z

Circle (presented as a triangle):

Dimension	Chains	Homomorphism	Cycles	Boundaries	Homology group
1	Z^3	(1,0,0) -> (1,-1,0); (0,1,0) -> (0,1,-1); (0,0,1) -> (-1,0,1)	(1,1,1)*Z	(0)	Z
0	Z^3	0	Z^3	{x+y+z=0}	Z???

See https://topospaces.subwiki.org/wiki/Homology_computation_for_spheres.

For any simplicial complex K:

• The 0-dimensional chain group is Z^p, where p is the number of vertexes in K.
• The 0-dimensional cycle subgroup is equal to the chain group (each chain is a cycle).
• Any two vertices joined by an edge yield homologous cycles.
• The 0-dimensional boundary subgroup are the p-tuples where, in each connected component, the sum is 0. So there are c constraints (where c is the number of connected components), and the rank of the boundary subgroup is p-c.
• The 0-dimensional homology group is Z^c, where c is the number of connected components. It is the free Abelian group generated by a set containing one vertex from each component.

Simplicial maps

Each simplicial map f between oriented simplicial complexes induces a chain map g between the corresponding chain complexes, via the formula:

• g(1*s) = 0, if dim(s) > dim(f(s))
• g(1*s) = 1*f(s), if dim(s) = dim(f(s)) and f on s is orientation-preserving;
• g(1*s) = - f(1*s), if dim(s) = dim(f(s)) and f on s is orientation-reversing.

See:

• Simplicial homology
• Simplicial approximation theorem

The Simplicial approximation theorem has a relative version.

The open star of a vertex v in a complex K, denoted ST^o(v,K), is the union of interiors of all simplices of K of which v is a vertex.

• It is an open subset of |K|.

The closed star of a vertex v in a complex K, denoted ST(v,K), is the union of closed simplices of K of which v is a vertex.

The intersection of the open stars of vertices is non-empty, if-and-only-if K contains a simplex spanning these vertices.

Induced homomorphisms of homology groups

Let G be a map between polyhedra |K|, |L|.

Let g be a map between the simplicial complexes K, L which are the triangulations of the above polyhedra.

By the simplicial approximation theorem, we can find such a g that is homotopic to G.

The map g defines a chain map from the chain-complex of K to the chain-complex of L. Hence, it defines a sequence of homomorphisms between the homology-groups of K and the homology-groups of L (for each n, g* maps H_n(K) to H_n(L)).

We define the homomorphisms G* as equal to g*; they do not depend on the selection of g. I.e, for every simplicial approximation g, we get the same g*.

In general, if f,g are homotopic, then their induced homomorphisms are identical for all n.

Degrees of maps between manifolds

Types of manifolds:

• smooth manifold - has an atlas in which all changes of charts are diffeomorphisms.
  • smooth orientable manifold - has an atlas in which all changes of charts preserve the orientation of $\mathbb {R} ^{n}$
• piecewise-linear manifold - has an atlas in which all changes of charts are simplicial maps (with respect to some triangulation).
  • piecewise-linear orientable manifold - has an atlas in which all changes of charts preserve the orientation of all simplices of dimension n.

A triangulated manifold of dimension n is orientable iff all its n-dimensional simplices can be oriented in a coherent way - i.e, in a way that respects the Doubly-Induced Orientation Theorem.

Theorem 9. For any closed triangulated manifold M of dimension n, the group H_n(M) is isomorphic to Z^k, where k is the number of connected components that are orientable.

In particular, if M is connected, then

• H_n(M) = Z if M is orientable,
• H_n(M) = trivial if M is not orientable.

(the boundaries-group is trivial; the cycles-group is generated by the sum of all coherently-oriented simplices of maximal dimension).

Let M1,M2 be closed connected oriented manifolds of the same dimension n. Then, H_n(M1) and H_n(M2) are both isomorphic to Z. Let f be a function from M1 to M2. It induces a homomorphism f* from H_n(M1) to H_n(M2), i.e, from Z to Z. The integer f*(1) is called the degree of f.

Geometric interpretation:

• The map f (or a simplicial approximation of it) maps each n-simplex in M1 to an n-simplex in M2.
• So the pre-image of each n-simplex s in M2 is some set of n-simplexes s1,...,sm in M1.
• Assign to each si the coefficient +1 if f preserves its orientation and -1 if f reverses its orientation.
• The degree of f is the sum of the coefficients. This degree is the same regardless of what target-simplex s we start from.

Geometric interpretation without simplices:

• The map f maps each point in M1 to a point in M2.
• Sard theorem implies that the set of singular values is small, in particular, there exists a regular value.
• Let s be a regular value on M2. Then, the inverse function theorem implies that its pre-image consists of a finite number of regular points s1,...,sm on M1.
• Assign to each si the coefficient +1 if the determinant of the Jacobian at that point is positive and -1 if it is negative.
• The smooth degree of f is the sum of the coefficients. This degree is the same regardless of what regular point s we start from.

The smooth degree always equals the degree; this can be proved using the category of the piecewise-linear manifolds.

The smooth degrees of homotopic smooth maps are equal.

Applications of the degree of a map

Maps from circle to circle

• The degree equals the winding number.
• deg(f)=deg(g) iff f is homotopic to g.

immersions from circle to plane

• Writhe number: take an immersion f on a circle. To each point in the circle, assign the endpoint of the unit vector f'/f. The result is a map from circle to circle. The writhe of f is the degree of that map.
• Whitney embedding theorem: For any n, there exists an immersion with writhe number n.
• Two immersions from the circle to R^2 are regular homotopic iff they have the same writhe number.
• Two immersions from the circle to S^2 are regular homotopic iff their writhe number has the same parity (???).

‎The number of common tangent lines of two immersed circles

Two circles are immersed in complementary half-planes. How many common tangent lines do they have?

• If the images are also circles, then the answer is 4.
• Otherwise, if the tangent lines are counted correctly (with correct multiplicity and sign), the answer is: 4*writhe(f)*writhe(g).

Relative homology

Given two chain-complexes K,L, the relative chain group C_n(K,L) is the chain group generated by all simplices with interiors in K\L. From this, it is easy to define the relative homology groups.

The chain complex of X U Y relative to Y is identical to the chain complex of X relative to X \cap Y.

The exact homology sequence

The sequence of chain groups is an exact sequence, if-and-only if all homology groups are trivial.

So, the homology groups of a chain complex provide a measure of its inexactness.

If L is a sub-simplicial-complex of K, then the following sequence is exact for any n:

0 -----> C_n(L) ---i--> C_n(K) ---p--> C_n(K,L) ----> 0

Where:

• i is the inclusion homomorphism - it is induced by the embedding of L into K.
• p is obtained by forgetting (=mapping to zero?) all simplices are K that are contained in L.

This sequence shows how a simplicial complex K can be decomposed to two pieces: L and K\L.

It is possible to construct from this short sequence, the following long exact sequence:

... ----> H_n(L) ---i*---> H_n(K) ---p*---> H_n(K,L) ---d---> H_{n-1}(L) ----> ...

where d = ${i^{*}}^{-1}\delta _{n}{p^{*}}^{-1}$

(this is a special case of the zig-zag lemma).

EXAMPLES:

• K = S^n, L = B^n (a ball embedded in the sphere).
• K = B^n, L = S^{n-1} (a sphere that is the ball's boundary).

By constructing the long exact sequences for each of these pairs, we can calculate the homology groups of an n-dimensional sphere:

• H_0 = H_n = Z,
• All other homology groups are trivial.

Another corollary is the Mayer–Vietoris sequence.

Axiomatic point of view on homology

There are various homology theories. Each homology theory is a functor:

• from the category of pairs of polyhedra,
• to the category of sequences of Abelian groups.

It must satisfy the four Eilenberg–Steenrod axioms: homotopy, long exactness, excision, dimension.

Uniqueness theorem: every functor that satisfies these four axioms is equivalent to simplicial homology (i.e, the homology groups generated by these two functors are isomorphic).

• Proof: the dimension axiom implies that the groups of a point are the same. Then, we can construct spheres and balls of higher and higher dimensions. Then, we can construct simplicial complexes by attaching new simplices and using the Five lemma (which is proved by diagram chasing).

Digression to the theory of Abelian groups

Any Abelian group can be presented by its generators and the relations on them. These relations can be presented in matrix form. For example, the matrix:

2 3 -1

0 1 2

1 3 0

1 4 2

represents the following Abelian group (a subgroup of Z^3):

{ (a,b,c) | 2a+3b-c=0 and b+2c=0 and a+3b=0 and a+4b+2c=0 }

which is equivalent to the quotient of Z^3 by the subgroup generated by the four elements: (2a+3b-c, b+2c, a+3b, a+4b+2c).

It is possible to bring each matrix to diagonal form using the 6 elementary operations: adding one row to another one, permuting the rows, changing the sign of a row, and the same for columns. So the above example can be brought to:

1 0 0

0 1 0

0 0 5

0 0 0

This means that the group is equivalent to Z_1*Z_1*Z_5, which is equivalent to Z_5.

It is also possible to bring each matrix to a canonical form = diagonal + each diagonal element is divisible by all preceding diagonal elements.

When the relation matrix is square, the absolute value of its determinant is either equal either to the order of the group (of it is finite) or to 0 (if it is infinite).

Calculation of homology groups

We are given a chain-complex in which all chain-groups are free and finitely-generated (like that of a simplicial complex). Let r_n be the number of generators of chain-group C_n.

Then, each boundary homomorphism d_n: C_n ---> C_{n-1} can be represented by a matrix A_n, with r_{n-1} rows and r_{n} columns: each column represents the image of one generator of C_n, in terms of the generators of C_{n-1}.

Since the boundary of a boundary is empty, d_n*d_{n+1}=0, so the product A_{n}*A_{n+1} = 0 = zero matrix with r_{n-1} rows and r_{n+1} columns.

We can simplify the matrixes with the following operations:

Operation on the basis of C_n	Operation on A_n	Operation on A_{n+1}
Replacing the generator x by x+y	Replacing a column x by x+y	Replacing a row y by y-x
Replacing the generator x by -x	Replacing a column x by -x	Replacing a row x by -x
Permuting two generators x,y	Permuting two columns x,y	Permuting two rows x,y

Using these elementary operations, it is possible to bring A_n to diagonal form, where the first k elements of the diagonal are nonzero and the rest are zero (these operations also change A_{n+1}).

Then, because A_n A_{n+1} = 0, necessarily the first k rows of A_{n+1} are zero.

Remove these first k rows and bring A_{n+1} to diagonal form (these operations also change A_{n+2}). Continue like this until all matrices are in diagonal form.

Then, the homology group H_n is read from the diagonal. It is the direct sum of the groups Z_{a11} * ... * Z_{akk} * Z^s, where s is the number of zero elements in the diagonal.

This also proves that:

• Any chain complex whose chain groups are all free, have finite ranks, and all trivial in negative dimensions, is isomorphic to a direct sum of elementary chain complexes - E(m) and D(m,k).
• Any sequence of finitely generated Abelian groups can be realized as a sequence of the homology groups of some free chain complex.

Cellular homology

To calculate the homology groups of a torus, we have to triangulate it, and the smallest triangulation has 14 triangles.

Alternatively, we can use singular triangulation, with 2 triangles.

Alternatively, we can use a cell complex, and then the torus is a single cell (a single square).

Cellular homology aims to use the most economic decomposition of a polyhedron into simple pieces.

The chain groups of a cellular complex are a generalization of those for simplicial complex. Now, the incidence coefficients can be arbitrary integers - not only -1,0,1. For example, the Klein bottle can be represented as a cell complex with:

• one vertex (0-dimensional cell);
• two loops from the vertex to itself (1-dimensional cells) marked by 1,2;
• one 2-dimensional cell whose boundary passes along the edges according to the rule {1,2,-1,2}.

There is an algorithm for computing H_1 of a cell complex, using a spanning tree of the 1-dimensional skeleton. Using it on the Klein bottle gives a matrix with diagonal (2,0), which corresponds to the group Z_2*Z.

For any polyhedron presented as a cell complex, its cellular homology groups coincide with the simplicial ones. This can be proved by the uniqueness theorem.

A third kind of homology is the singular homology. Again the homology groups are the same. Many theorems become much simpler. However, there are infinitely many singular-simplices, and the calculation of the homology groups is harder.

Lefschetz fixed-point theorem

The trace of an endomorphism is the trace of the matrix that represents the endomorphism on the free part of the group (ignoring the periodic parts).

The trace is additive: the trace of an endomorphism on the direct sum of two groups is the sum of the traces of the induced endomorphisms on each of the groups.

The Lefschetz number of an endomorphism-complex on a chain-complex is the alternating sum of the traces of the each of the individual endomorphisms.

A map from a simplicial-complex K to itself induces an endomorphism-complex on C(K), so it also has a Lefschetz number.

The Lefschetz number of the identity map from K onto itself equals the Euler characteristic of K.

The Lefschetz number of a map f that equals the identity on each invariant simplex, equals the Euler characteristic of the subcomplex of K that corresponds to the fixed-point-set of f.

The homological Lefschetz number of an endomorphism-complex is the alternating sum of the traces of the individual endomorphisms induced on the homology groups. It is equal to the Lefschetz number.

Corollary: The Euler characteristic of a finite simplicial complex K equals the alternating sum of the ranks of the homology groups of K.

An arbitrary map f from a polyhedron to itself also has a Lefschetz number - the alternating sum of the traces of the endomorphisms on its homology groups.

If f is simplicial, then its Lefschetz number equals the Euler characteristic of the fixed-point-set of f. Examples:

• The symmetry of S^2 about its center - its Lefschetz is 0; it fixed-point-set is empty so its Euler is 0 too.
• The symmetry of S^2 about its diameter axis - its Lefschetz is 2; its fixed-point-set contains two points (the poles) whose Euler is 2 too.
• The symmetry of S^2 about its equatorial plane - its Lefschetz is 0; its fixed-point-set is the equatorial (a circle) whose Euler is 0 too.

If f is not simplicial, then this equality is not guaranteed. However, it is still true that: if the Lefschetz number is nonzero, then f has at least one fixed point.

Homology with coefficients

We can construct the chain-groups, instead of integer coefficients, with coefficients in any Abelian group. This can be more convenient in some cases. E.g, if we take the coefficients from the Z_2, then -1 and +1 coincide, we do not need to track the orientation of simplices, and can consider unoriented complexes.

A cycle with coefficiencts in Z_2 is a collection of n-dimensional simplices such that each simplex of dimension n-1 is adjacent to an even number of them.

With coefficients in Q or R (or any field F with characteristic 0), there are no periodic groups ("torsion"), so each homology group is of the form F^k, i.e, it is completely determined by its rank.

Universal coefficient theorem: for each complex K and group G:

• the chain group C_n(K,G) is isomorphic to C_n(K,Z) tensor product G.
• the homology group H_n(K,G) equals H_n(K,Z) tensor product G direct sum H_{n-1}(K,Z) torsion product G.

Elements of cohomology theory

Cohomology is a contravariant functor.

Given a chain-complex, the n-th cochain group is the group of functionals from the n-th chain group to Z.

The coboundary homomorphisms map the cochain of dimension n-1 to the cochain of dimension n.

The chain group C_n is isomorphic to the cochain group C^n, but there is no natural isomorphism.

Calculation of cohomology groups is similar to homology groups - only the order is opposite.

The matrices of the coboundary homomorphisms are the transpose of the matrices of the boundary homomorphisms.

The cohomology group H^n(C) is isomorphic to the direct sum Free(H_n(C)) + Torsion(H_{n-1}(C)). This can be verified by decomposing C to the elementary components.

The direct sum, n=0 to infinity, of H^n(K) of an arbitrary complex K, possesses a natural ring structure, based on the cup product and cap product.

The tensor product of chain complexes is related to the cartesian product of the simplicial complexes.

The chain complex C(K)tensor productC(L) is isomorphic to C(K x L).

Elements of homotopy theory

Coverings

--- TO READ: Singular homology, Cobordism