Algebraic Topology
Simplical and Singular Homology
My first attempt at reading into algebraic topology was thwarted by the lack of abstract algebra and group theory background. The book on algebraic topology by Allen Hatcher is one of the often recommended ones, and considering the rigorous way it treats proofs, I can certainly see why. It's also my favorite kind of book when it comes to learning new stuff. Most of topology will have trouble avoiding the fundamental groups, but it's just not sensible to address the general fundamental group (i.e. the fundamental group for arbitrary n) without a background in group theory. Higher-number homotopy groups of the fundamental group however can't be computed directly, which introduces the need for a geometric understanding of higher-dimensional spaces. The study of these will often utilize the simplex, cells with an edge length of one in all directions, to construct objects in higher-dimensional space. For topology, where the exact geometry is secondary, the homology emerging from these simplexes is more interesting.
A mild generalization of a simplicial complex should be familiar to people who have heard topology: The Δ-Complex. Often this is used to construct the torus, the projective plane and the Klein bottle by taking a square and glueing the edges together in some prescribed direction. The Δ-complex takes this idea and applies it to the n-simplex, which is the smallest convex set in a Euclidean, n-dimensional space containing n + 1 points (not in a hyperplane, of course). The standard n-simplex is defined as
with unit vectors as edges. These are constructed such that removing one edge will leave us with an (n-1)-simplex. This is called a "face". The union of all the faces of an n-simplex is its boundary, the open simplex is its interior. The appropriate notation is adopted.
The somewhat familiar Δ-complex is a collection of maps
where n depends on α, so that:
by the canonical linear homeomorphism preserving the order of vertices. A ⊂ X is open, iff
X can then be built as a quotient space of disjoint simplices, which makes it being Hausdorff apparent through the homeomorphism of each restriction onto its image, defined to be an open simplex in X. With this, the free abelian group with basis of the open n-simplices e(n, α) of X can be defined. We write it as Δ(n, X), elements of which are called "n-chains", written as finite formal sums. over the n-simplices. For the general Δ-complex X, the "boundary homomorphism" is
The composition
vanishes. It opens the consideration of sequence of descending homomorphisms of abelian groups, where
These sequences are called "chain complexes", and their existences imply that
The n-th homology group of the chain complex is the quotient group
where the elements of the kernel are "cycles" and elements of the image are "boundaries". Elements of the quotient group are cosets of the Image and are considered homology classes, so that two cycles representing the same homology class are "homologous", i.e. they only differ in their boundaries. If C(n) = Δ(n, X), then the homology group is the n-th simplicial homology group of X.
Singular n-simplices are defined as a map ρ: Δ(n) → X, where ρ isn't required to be a nice (or singularity-free) embedding, provided that it's continuous. Elements of the free abelian group C(n, X) with a basis consisting of singular n-simplices, are singular n-chains with a boundary map defined as it is for simplical n-chains. Similarly, the singular homology group is constructed in the same way that the simplical ones are. For homeomorphic spaces, the singular homology groups are isomorphic, while it's not immediately clear whether singular homology groups can even be finitely generated due to the number of elements in the group. Through the construction by singular complexes, the singular homology could be considered a special case of the simplicial homology, luckily.
As the singular simplex can be thought of as having a path-connected image, through splitting the free abelian groups, the decomposition of a space into its path-components have an isomorphism of the singular homology group, constructed through the direct sum over the singular homology groups of its components. Nonempty and path connected spaces will have a 0-th singular homology group isomorphic to the group of whole numbers. If the space is a point, then in addition, the n-th homology group will be null for n > 0.
Now consider a map f: X → Y inducing a homomorphism f#: C(n, X) → C(n, Y), defined by composition of all singular n-simplicies ρ, and subsequent linear extension of g. Then,
Through tracing the squares in the resulting commutative diagram, it can be shown that f#∂ = ∂ f#. This also means that the maps f# define a chain map from the singular chain complex of X to the singular chain complex of Y, or, algebraically: Chain maps between chain complexes induce homomorphisms between their homology groups. Assuming two maps f, g: X → Y are homotopic, they induce the homomorphism f* = g*: H(n, X) → H(n, Y). With this, we can define the "chain homotopy" between chain maps f# and g# as some P for which ∂ P + P ∂ = g# - f#. Maps that are chain-homotopic induce the same homomorphism on their homology.
Assuming that X is a Δ-complex and A ⊂ X itself a (sub-)complex, then A is the Δ-complex constructed by any union of simplices of X. The relative groups are then defined parallel to the singular homology, i.e. by their relative chains. As a diagram, it will show up a long exact sequence with nodes consisting of simplicial homology groups, with a canonical option for the homomorphism between the singular and simplicial homology groups, induced completely through the chain map. For complex pairs (X, A) (and all n), these homomorphisms are also isomorphic. At this point, we've seen the 5-Lemma in the context of both analysis and category theory. Suffice it to say, that it comes back here, constructed using said long exact sequences, in the same way the general construction from category theory went. For this specifically, it follows that H(n, X) is finitely generated for Δ-complexes, provided it consists of a finite number of n-simplices. Writing it as the direct sum of its cyclic groups, then the number of ℤ summands is considered the n-th "Betti number" of X. The finite cyclic summands have an order, given by some integer, called the "torsion coefficient". We also tap what we know about Brouwer's Theorem regarding maps between n-dimensional spheres and what that can fix about homologies. The "degree" of a map between n-dimensional spheres f (with n > 0) is defined through the induced map f*, which is homomorphic between the singular homology groups (infinite cyclic groups, as we should know from algebra) of the form f*(α) = dα for some integer d, dependent only on f. Through parametrization of the n-dimensional sphere and construction of a homotopy, the degree can be used to show that the n-dimensional sphere has a continuous field of nonzero tangent vectors iff n is odd. Computing the degree of some map f can be done by construction of a commutative diagram linking the homology groups so that the homology groups of the n-dimensional sphere are limites in the category-theory sense. This of course has some consequences that involve splitting exact sequences, which are best illustrated using CW complexes instead of a generalized space. The lemmas of course don't fundamentally change. A special case of these long exact sequences are Mayer-Vietoris sequences. Take A, B ⊂ X : X = A° ∪ B°.
It's a good tool for induction arguments, if A, B and A \cap B are known.
Generalizing homology theory can be done using chains of the form Σ n ρ, where ρ is a singular n-simplex, and the coefficients n are taken to lie in a fixed abelian group G. They form an abelian group C(X; G), with C(X, A; G) = C(X; G) / C(A; G). The definition of the boundary maps is retained. The resulting homology groups are said to have "coefficients in G", with their reduced groups being defined through the augmented chain complex
where ε is the sum of coefficients. For a map f between k-dimensional spheres with degree m, f* between the homology groups is the multiplication by m.
Simplicial Approximation
Short Chapter this week, but not everything has to be a long-form project. From an entry-level topology course, the readers should be familiar with both the fundamental group and its relation to homologies, from which the Borsuk-Ulam Theorem is usually derived. These form the fundamentals for the "simplicial approximation theorem". It's used to bring spaces into a structure of simplicial complexes, which, while not as widely used as CW complexes, will still introduce a structure to the space which is familiar. Occasionally the homology lying over spaces constructed from simplicial will be beneficial also. For simplicial complexes K and L, a map f: K → L is considered simplicial, if it sends each simplex of K to a simplex of L by associating vertices to vertices linearly. The Simplicial Approximation Theorem states that f is homotopic to a map that is simplicial with respect to some iterated barycentric subdivision of K (subdivision, since f need not be surjective). We note St ξ for the star of a simplex ξ and st ξ for its open star (i.e. an open set with the closure St ξ). For vertices in a simplicial complex, the intersections of their open stars is non-empty, if a simplex ζ exists which contains all vertices. The set of all these vertices then (trivially) create a simplex ξ in ζ, meaning that the intersection of all open stars of all vertices is exactly st ξ. The Simplicial Approximation Theorem can be proven by choosing a metric K restricting the standard Euclidean metric on each simplex of K, and finding a Lebesgue number of an open cover of K, so that each simplex in the subdivision of K is less than half this number in diameter. The closed star of each vertix has a diameter less than the Lebesgue number. Mapping this with a map f, as constructed will map each vertex v to some g(v) of L, which will define g as a map that extends to a simplicial map g: K → L. If one wants the maps to preserve basepoints, g can be set so that it maps vertices that are already sent from K to L by f to themselves.
A homomorphism Ψ: ℤⁿ → ℤⁿ with the matrix [aᵢⱼ] and the standard trace operator. Conjugate matrices have the same trace, so tr Ψ is independent of the chosen basis. If Ψ is a homomorphism of a finitely generated abelian group A, then tr Ψ is the trace of the induced homomorphism φ: A/Torsion→A/Torsion. Applying this to CW-complexes gives rise to a new quantity, the "Lefschetz number"
If f is the identity (or homotopic to it), then τ(f) is the Euler characteristic. From this emerges the Lefschetz fixed point theorem, stating that if X is a finite simplicial complex (or any retract of a finite simplicial complex), and f: X → X is a map with τ(f) ≠ 0, then f has a fixed point.
To see this, note that every compact, locally contractible space, for which an n exists, so that the space can be embedded into ℝⁿ, is a retract of a finite simplicial complex.
Having seen a method to approximate continuous maps into homotopic simplicial maps, a similar method for spaces should be possible. There are at least spaces, which are homotopy-equivalent to simplicial complexes. The most common class of spaces in algebraic topology are CW complexes, so it makes sense to start there. Every CW complex X is homotopy equivalent to a simplicial complex, chosen to be of the same dimension. The proof can be approached inductively, by showing the statement for unions of subcomplexes, homotopy equivalent to the skeleta of X. The inductive step will introduce an (n+1)-cell of X, attached by a map φ. A map corresponding to φ under homotopy equivalence is homotopic to a simplicial map by the simplicial approximation theorem. The union of the subcomplexes and the (n+1)-cell is homotopy equivalent to the skeleton of X and (n+1)-cell. It can be helpful to construct simplicial analogs of mapping cones to see the last step, instead of going through the mapping cone directly. It really depends on how comfortable one is with category theory.
Cohomology Groups
In the usual way that we define co-anythings (or "contravariance", if you wanna be all fancy and correct), the cohomology is the dual analogue to the homology. Where this is often all one needs to completely define an object, one finds, that in the case of the cohomology, a new type of product spring up. The "Cup Product" forces the homology group of a space into a ring, and since rings are a very comfortable structure to work within, exploiting this product is very much of interest. The covariant product in the homology is the very familiar cross product with
for X, Y CW-complexes. However, in general it's not always clear whether X ⨉ X even exists for the general case. Occasionally, the only natural maps for which X ⨉ X → X is a projection map, which, as homomorphic 2-forms, collapse one of the factor spaces into a point, and leaving only trivial homologies. The cross product of the cohomology turns this around, so the interesting map becomes X → X ⨉ X, which is always possible through the diagonal map, which is well-behaved in homologies.
The cohomology also have another set of "cohomology operations", which are a bit more complicated to construct, and which we'll get back to another time.
First, we'll want to get to cohomology groups. The general chain complex of free abelian groups runs as follows:
and the dualization of this chain flips the arrows and replaces each group in the chain with its dual "cochain group" C = Hom(C, G) where C → G. The boundary map is replaced by the dual coboundary map. This definition is as we should expect. The homomorphism α: A → B has the dual homomorphism α*: Hom(B, G) → Hom(A, G), α*(g) = gα, sending B → G to A → B → G. Because of the algebra on dual homomorphisms, from the boundary map condition ∂∂ = 0, it follows that the coboundary map has the same condition δδ = 0. In a way, the cohomology group H*(C; G) is the homology group Ker δ / Im δ at C* in the cochain complex. This also means, that such a cohomology group is only determined by G and the homology groups H(C). The proof of this is lengthy, and can be found through splitting of a short exact sequence and looking at the Kernel of the homomorphism from H(C; G) to Hom(H(C), G). Given two parallel chains of free resolutions F, F' of the abelian groups H, H', then the homomorphism α: H → H' can be extended to the pairs of free resolutions F → F', so that the resulting chain maps are chain homotopic. For any two free resolutions F, F' of H, there are canonical isomorphisms Hⁿ (F; G) ≈ Hⁿ(F'; G) ∀ n.
Every abelian group H has free resolutions, so that 0 → F' → F →H → 0 by first choosing a set of generators for H and having the basis of F in one-to-one correspondence with said generators. This spans a surjective homomorphism f: F → H between basis elements and (a subset of) generators. Since f is a subgroup of a free abelian group, it's kernel is free, and the with inclusion g: F' → F one can just define F' to be the kernel of f. Obviously, from the previous statement, Hⁿ(F;G) = 0 for all free resolutions up the chain than F', and for n = 0, that is the same. Only H(F'; G) then is interesting. Then, from a chain complex C of free abelian groups with homology groups H(C) implies cohomology groups, determined by split exact sequences
This is the universal coefficient theorem for cohomology, which, combined with the five-lemma, shows that a chain map between chain complexes of free abelian groups inducing isomorphy on homology groups implies isomorphy between cohomology groups with any coefficient group.
On the topological avenue, too, we require dual definitions of objects defined previously. In this way, the singular chain group is defined against the "singular n-cochains with coefficients in G" Hom(C(X), G) so that n-cochain φ ∈ C(X; G) is assigned a value in G for each singular n-simplex. With the coboundary map and cohomology group, a cochain complex emerges from which becomes clear that elements of ker δ are cocycles, and those of Im δ are coboundaries. This means that for a cochain φ: δφ = φ∂ = 0. It vanishes on boundaries. The definitions that were derived from homology theory can be applied to these basics without problems, though the naming convention persists, marking the covariance for legibility.
Cup Product
We return to the cup product, that was, after all, a central feature of the homology covariance. At the start of the previous week's entry, the cup product was defined with a relatively lengthy preamble. Now having defined the basics of cohomologies, it can be done much faster. A cohomology with coefficients in a ring R and cochains φ ∈ Cⁿ(X; R), Ψ ∈ Cᵐ(X; R), has a cup product φ ∪ Ψ ∈ Cⁿ ᐩ ᵐ(X; R) is the cochain with values on a singular simplex σ: Δⁿ ᐩ ᵐ → X
It's related to the coboundary map by
It seems then, that the cup product enjoys linearity, so the cup product of two cocycles gives another cocycle. The cup product of cocycles and coboundaries will result in a coboundary, since the other component cancels due to the coboundary condition. This opens the possibility of an induced cup product mapping
which is actually a quite simple object. It's associative and distributive, and if R has identity, then so does the cup product. This is the class which lives in the singular homology group for n = 0, so defined by the 0-cocycle taking the value 1 on all singular 0-simplicies. For simplicial cohomology, this construction works the same way. From the cochain formula, it can be inferred that the induced maps satisfy f*(α ∪ β) = f*(α) ∪ f*(β). Define the cross product.
Whether the cup product is equivalent to the multiplication in a ring structure on the cohomology groups of a space is interesting, considering that it's already associative and distributive. The naive approach is to find groups, so that
If identity is given, and the space these live on are rings, then the cup product gives rise to an R-algebra. This covariance doesn't really have any topological significance, as the cohomology can be treated as a "graded ring" (i.e. as having a decomposition as a (direct) sum of additive subgroups, such that
A generalization of graded rings has the following property:
By this mechanism, cup products can distinguish infinitely many different homotopy classes of maps
meaning, they are closely related to Hopf invariance. For a commutative R:
A commutative, graded ring with with this property is usually referred to as simply "commutative" in algebraic topology, though the literature will describe these as "graded commutative", "anticommutative" or "skew commutative" to avoid confusion.
The cross product in this context can be considered an "external cup product" H*(X;R) ⨉ H*(Y;R) → H*(X ⨉ Y;R) with a ⨉ b = p(a) ∪ q(b) where p, q are projections of X ⨉ Y onto X and Y. By replacing the direct product with a tensor product, this construction can be given homomorphism, and hence isomorphisms. Let X, Y be CW complexes, and Hᵏ(Y;R) a finitely generated free R-module, then H*(X;R) ⨂ H*(Y;R) → H*(X⨉X;R) is an isomorphy of rings. Such isomorphies are also called "Künneth formulas". With the Hopf theorem, Künneth formulas can help show that if the n-dimensional space of real numbers has the structure of a division algebra over its scalar field, then n is a power of 2.
Poincare Duality
For the following, the definition and properties of manifolds are presupposed. For closed, orientable manifolds M of dimension n, Poincare duality defines isomorphisms
Convention usually has (co-)homology groups of negative dimension are null. This opens a symmetry in the homology of closed orientable manifolds. The proof of this duality can be done through the geometry of dual cell structures, by constructing objects out of polyhedric cell structures. Most topologically interesting is the construction of diverse tori, which will have different geni, depending on the base shape of the cell. Given dual cell structures C, C* on a closed surface M, the possible pairing of such structures are
By writing the assingment as a tensor, the boundary map between cells becomes the cellular coboundary map by the cyclicality. By the Poincare duality,
The orientability of M takes care of the sign ambiguity. This formalism can be extended into arbitrarily high dimensions. The orientation of n-dimensional real spaces at a given point is technically ambiguous, but can be fixed by convention. Usually, this is by the choice of generator of the infinite cyclic group. Of course by this definition, the orientation of one point infers that of all other points in that space. Formally, this is done by canonical isomorphisms. This means that manifolds of all dimensions can be oriented. Such a choice of generator is called the "local orientation" at a point. If M is connected, then it's orientable iff its orientable two-sheeted covering space has exactly two components. It's also simply-connected. Closed connected n-manifolds are orientable, if the map H
is isomorphic for all x in M. If they're not, then the map is injective with {r ∈ R | 2r = 0} for all x ∈ M. For i > 0, the map evaluates to zero. For R = ℤ, the map maps onto ℤ itself (or 0, if it's not orientable). In either case for R = ℤ/2 it maps onto ℤ/2. An element of H(M;R) with an image in H(M|x;R) that is a generator ∀ x is a "fundamental class" for M with coefficients in R. These can only exist for closed, R-orientable M. They can also be referred to as an "orientable class" for M. Closed n-manifolds with the structure of a Δ-complex can form the basis for a more direct construction construction for a fundamental class. For ℤ coefficients, through the representation of the fundamental class and the mapping of the fundamental class to a generator of H(M|x;R) for the interiors of σ so that the linear combination of the fundamental class' representation only carries coefficients of ±1. Then, if two σ share a common (n-1)-dimensional face, their coefficients are codependent. The representation can then only be a cycle if M is orientable, defining a fundamental class. If M is a connected noncompact n-manifold, then
To define Poincare duality, first define an R-bilinear cap product for an arbitrary space X and coefficient ring R
which induces a cap product in (co-)homology, by
Proof follows through direct calculation. Its properties induce a cap product that is R-linear in each variable. For a closed, R-orientable n-manifold M, if
For such M, a duality map can be defined by a limiting process. In that way,
is isomorphic for all k. The cup and cap product are related by
For the cup product structure of manifolds, a closed R-orientable n-manifold with the pairing
These are nonsingular, if A → Hom(B,R), B → Hom(A,R) are both isomorphisms. The cup product pairing is nonsingular for closed R-orientable manifolds, if R is a field, or R = ℤ with torsion in H*(M;ℤ) is factored out.
Universal Coefficients for Homology
In the discussion around cohomology, the concept of universal coefficients have arisen naturally, and this concept can be turned around for homologies as well. It goes back to the definition of homology with coefficients, but expressed through tensor products. The chain group Cₙ(X;G) was defined as consisting of finite formal sums Σgₙσₙ with gₙ ∈ G, σₙ: Δᵏ → X, meaning it's a direct sum of copies of G with a copy for each singular n-simplex in X. The relative chain group Cₙ(X, A;G) then must also be a direct sum of copies of G with one for each singular n-simplex in X, which is not also in A. By Kuenneth, Cₙ(X, A;G) is naturally isomorphic to Cₙ(X,A) ⨂ G by Σgσ → Σσ ⨂ g so that the boundary map for the relative chain group becomes ∂⨂1: Cₙ(X,A) ⨂ G → Cₙ₋₁(X, A) ⨂ G with the usual boundary map for coefficients in ℤ. The algebraic problem of computing the homology groups out of a chain complex of free abelian groups can be approached by understanding a similar process applied to subprocesses consisting of cycles and boundaries within free abelian groups. For this, take Z(n) = Ker ∂ₙ ⊂ Cₙ, B(n) = Im ∂ₙ₊₁ ⊂ Cₙ where the boundary function is without restrictions within the subgroups. The resulting commutative diagram consists of two rows, splitting over B(n - 1), B(n - 2), so Cₙ ≈ Z(n) ⊕ B(n - 1). The same can't quite be said of the chain complex C, due to nontriviality of their boundary maps in C. However, returning to the original problem (by tensoring everything with G) retains the splitting of either row, which gives rise to the long exact sequence of homology groups. The boundary maps within this sequence can be defined as inclusions by "finding" the pull-backs in the diagram. By these inclusions, their co-kernels can be used to "shorten" the sequence, where the byproduct is, that the co-kernel is exactly Hₙ(C) ⨂ G. This, along with the fact that for abelian groups A, B, C with i: A → B, j: B → C → 0, builds an exact sequence, this sequence is still exact when tensoring the groups with G, and the functions with identity, will help define canonical isomorphisms between certain types of homology groups. A free resolution, too, is an exact sequence of free groups F(n) of an abelian group H, each abstracted away from H by some function fₙ. Any two such sequence of the same group have a canonical isomorphism between their homology groups for all n.
Groups Hₙ(F ⨂ G) only dependent on H and G is noted as Torₙ(H, G). The question is, whether it exists. One can be certain that Tor(H, G) is a functor, so that a: H → H', b: G → G' with homomorphisms a*: Tor(H, G) → Tor(H', G), b*: Tor(H, G) → Tor(H, G'). To find Tor(H, G) explicitly, the universal coefficient theorem for homology is going to be helpful. Chain complexes of free abelian groups induce short exact sequences 0 → Hₙ(C) ⨂ G → Hₙ(C, G) → Tor(Hₙ₋₁(C), G) → 0 ∀ n, G. There exists a way to get the sequence to split. This also exists for C being any pair of spaces. In this case, the split is not natural, since it would have to induce trivial maps between homology groups. The sequences themselves though are natural with respect to maps between pairs of spaces. From these ideas, a few statements can be generally assumed. Note T(A) for the torsion group of A.
1. Tor(A, B) ≈ Tor(B, A)
2. Tor (⊕ₙ Aₙ, B) ≈ ⊕ₙTor (Aₙ, B)
3. A, B (torsion-)free ⇒ Tor(A, B) = 0
4. Tor(A, B) ≈ Tor(T(A), B)
5. Tor(ℤₙ, A) ≈ Ker(A → A)
6. ∀ short exact sequence 0 → B → C → D → 0 ∃ naturally associated exact sequence 0→ Tor(A, B) → Tor(A, C) → Tor(A, D) → A ⊕ B → A ⊕ C → A ⊕ D → 0.
From these rules follows directly that Tor(ℤₙ, ℤₘ) ≈ ℤₛ where s is the gcd of n and m. This makes Tor(ℤₙ, ℤₘ) isomorphic with ℤₙ ⨂ ℤₘ. It follows, that for finitely generated groups A, B: Tor(A, B) ≈ Tor(T(A), T(B)). This, along with 3. and 4. is (roughly) where 'Tor' takes its name from.
Hₙ(X; ℚ) ≈ Hₙ(X; ℤ) ⨂ ℚ. For finitely generated Hₙ(X; ℤ), dim[Hₙ(X; ℚ)] = rank[Hₙ(X; ℤ)]. If both Hₙ(X; ℤ) and Hₙ₋₁(X; ℤ) are finitely generated, then for a prime p, Hₙ(X; ℤₚ) consists of ℤₚ summands for all ℤₛ summand in Hₙ(X; ℤ), Hₙ₋₁(X; ℤ), where s is a positive exponent of p.
H-Spaces, Hopf Algebras
First, define a H(opf)-Space as a space X with a continuous multiplication map μ: X ⨉ X → X and an identity element e so that x → μ(x, e), x → μ(e, x) are homotopic through map identity. Classical examples of H-spaces are topological groups. Topologically, some of the simplest H-spaces are unit spheres, specifically S1 in ℂ, S3 in the quaternions and S7 in octonions, as the multiplications on these spheres are continuous in the division algebras. Since all vectors in these spaces have the length 1, the norm is preserved and the the identity is included. This, along with continuity of operation qualifies S1 and S3 as Lie groups. Octonions aren't associative, so S7 is not a group at all. Actually, S0, S1, S3, S7 are the only spheres that are H-spaces. This highlights though, how H-Spaces are fundamentally different objects from groups. There exist H-spaces, which are associative, but not inverse (ℂP∞).
Starting off with a commutative ring R as coefficient allows the consideration of the cohomology ring H*(X; R) as an algebra over R. If X is an H-space which is path-connected (so the zero-th homology group is isomorphic to R), and the cohomology groups are finitely generated for their orders n, so that their cross product form isomorphisms.
To define Cohomologies in Hopf Algebras, we keep in mind the following two properties of any H-space X
1. X is path-connected, meaning H⁰(X; R) is isomorphic to R
2. Hⁿ(X; R) is a finitely generated free R-module ∀ n. The cross-product of cohomology rings is then also an isomorphism.
The product X⨉X → X then induces one between the cohomology rings from X⨉X to X. Then, Δ: H*(X;R) → H*(X;R) ⨂ᴿ H*(X;R) is an algebra homomorphism and
A Hopf-algebra then is a graded algebra A = ⊕ₙ Aⁿ with n ≥ 0 with an identity element 1 in A⁰ so that R → A⁰, r → r·1 isomorphic (i.e. A is connected), and A has a diagonal. Note, that the tensor product of Hopf algebras is a Hopf algebra itself, with coproducts Δ(a⨂b) = Δ(a) ⨂ Δ(b). Commutative and associative Hopf algebras over fields F of characteristic 0, so that Aⁿ is finite-dimensional over F for all n are isomorphic as algebras. For exterior algebras on odd-dimensional generators, it's as an algebra to the tensor product, and for exterior algebras on even-dimensional generators, it's as a polynomial algebra.
Homology groups gain a special product, the pontryagin product, on H-spaces. With the previous nomenclature, and ⨉: H⁎(X;R)⨂H⁎(X;R) → H⁎(X⨉X;R), μ⁎: H⁎(X⨉X;R) → H⁎(X;R), is the composition ⨉(μ⁎). It's then a family of not generally associative or commutative bilinear maps. For CW-complexes X and cellular maps μ, the pontryagin product can be found through the cellular chain map concatenating the indices of the chain homologies with the product. This product can be used to define a dual construction to the Hopf algebra. A H-space X with finite-dimensional homology groups over a field R for all indices n. It's identical to the Homomorphism Group of the Hₙ(X;R) with the coproduct as defined. This coproduct is dual to the pontryagin product, both induced by the H-space product μ. If the Hopf algebra A over a finitely generated, free R-module in all dimensions R has a product and coproduct that both have duals, these induce a dual Hopf space, which itself is a Hopf algebra.
Homotopy Groups
In the topology course I heard, we did most of our proofs at the end using the homotopy groups, and heavily relying on Seifert-van-Kampen, which helped to establish something akin to a full theory regarding the use of homotopies. However, at the time cohomologies weren't introduced yet, and the connections between cohomologies and CW complexes, which lie at the root of homotopy theory, will likely expand the scope this theory.
The most basic constructions when it comes to CW complexes will invariably be based in unit cubes of some kind. In these constructions, the homotopy group is the set of homotopy classes of maps from the cube and its boundary in n dimensions onto a space with a basepoint. From topology we can assume that the sum built from composition is well defined. Assume f, g to be maps within these homotopy classes. In the logic of homotopies, the domains of f and g can be shrunk down up to the boundaries of f and g, by mapping everything outside the boundary to the basepoint of the space. The resulting "cubes" of f and g can be moved around one another, as long as they remain disjoint. None of these operations change the nature of the homotopy. Since the n-Spheres can be obtained from unit cubes by removing the boundaries, the same basic idea is true for n-Spheres. The sum however needs to be linked through the basepoint, hence, why n-spheres are always combined using the wedge product. This makes the basepoint a central characteristic of the space, depending on whether every body within the space can be positioned arbitrarily with respect to it. This condition is synonymous with path-connectedness. From topology we should know the problems that arise when loops are allowed to exist in this space. Moving the basepoint is formalized using a "change of basepoint transformation" B. If a loop encloses the basepoint, and B moves the basepoint beyond the confines of this loop, then the loop must either be transformed alongside it, or the two spaces are not isomorphic, which is contradictory to the definition of isomorphic transformations. The homological classes of loops for which this is a problem can be mapped to a class of valid transformations, resulting in a homomorphism from the homotopy group to the group of Automorphisms on the homotopy group, called the "action". Each element links maps on the space with a loop through composition, inducing a module over the group ring (with appropriate dimension). If the action is trivial (a property of the space), then the space is considered "n-simple", occasionally also as "abelian space" or "nilpotent".
In this way, the n-homotopy group behaves as a functor, which creates a homotopy equivalence between basepointed spaces, as long as f: (X, x) → (Y, y) with f ⁎: πₙ(X, x) → πₙ(Y, y) so that f([g]) = [fg]. For covering spaces, projections then automatically induce such isomorphisms for all n > 1.
A relative homotopy group is a generalization for a basepointed space. Take the face of an n-dimensional volume with the last coordinate s = 0, along with the closure of the difference between this face and the boundary of the volume (or more concise: the union of all remaining faces). The relative homotopy group for n > 0 then is the set of homotopy classes of maps
The case for n = 0, doesn't seem to have a proper/unique definition. Since π(X, z, z) = π(X, z), absolute homotopy groups are a special case of relative homotopy groups. Relative homotopy groups of dimension 3 and above are abelian. Relative homotopy groups can be fit into a long exact sequence
where i, j are inclusions. The boundary map is homomorphic for n > 1. Even near the end of the sequence, where groups structures have been stripped away by inclusions, the exactness still holds and is well defined. For intuition, the images of the maps for the kernel of the next.
Whitehead's theorem deals with homotopy groups of CW complexes: If a map f: X → Y between connected CW complexes induces isomorphisms f⁎: πₙ(X) →πₙ(Y) ∀ n, then f is a homotopy equivalence. If f is the inclusion of a subcomplex, then X is a deformation retract of Y. Note, that X and Y that have isomorphic homotopy groups, don't necessarily have a map inducing isomorphisms on homotopy groups.
A map between CW complexes so that f(Xⁿ) ⊂ Yⁿ for all n is called a "cellular map". In fact, by the cellular approximation theorem, every map f: X → Y between CW complexes is homotopic to a cellular map. Cellular maps f on subcomplexes A ⊂ X, the homotopy may be taken to be stationary on A. Maps from n-cubes to a space constructed through attaching a cell eᵏ to a subspace of W are homotopic to a map g in which a simplex in eᵏ exists, on which g is the restriction of a linear surjection.
A "weak homotopy equivalence" induces isomorphisms πₙ(X, z) → πₙ(X, f(z)) for all n ≥ 0, and all choices of basepoint. As per Whitehead, weak homotopy equivalences between CW complexes are homotopy equivalences, but in general, weak homotopy equivalence is strictly weaker than homotopy equivalence. Every space can be assigned a CW complex and a weak homotopy equivalence mapping onto it. This map is called a CW approximation.
A pair (X, A) where A is a nonempty CW within X, an n-connected CW model for (X, A) is an n-connected CW pair (Z, A) along with a map f: Z → X and identity f|A so that f⁎: πₘ(Z) → πₘ(X) is an isomorphism for m > n, and an injection for m = n, regardless of basepoint. A similar constriction mapping from the homotopy groups of A to those of Z are isomorphisms for m < n and surjections for m = n. By factoring, Z can act like a hybrid of A and X for homotopic considerations, though it tends towards A with increasing n. This means that, with the correct choice, one can get Z by attaching cells of dimension greater than n to A. Assuming (X, A) is an n-connected CW pair, then there is a relation (Z, A) isomorphic to (X, A) rel A with Z - A of dimension greater than n. n-connected CW models for (X, A) are unique up to homotopy equivalence rel A. Weak homotopy equivalice f: X → Y induces isomorphisms f⁎: Hₙ(X;G) → Hₙ(X;G) and f*: Hⁿ(X;G) → Hⁿ(X;G) for all dimensions and coefficient groups G.
Elementary Methods of Calculation
Homotopy groups, in contrast to homology groups, don't have a general excision property. Theirs is confined to a dimension range, depending on connectivities. For CW complexes X, decomposing into a union of subcomplexes A, B with some nonempty connected intersection. For m-connected (A, C) and n-connected (B, C), the map between their i-th homotopy groups induced by inclusion is an isomorphism for i < m + n, and a surjection for i = m + n. From this follows the "Freudenthal suspension theorem", by which the suspension πᵢ(Sⁿ) → πᵢ₊₁(Sⁿ⁺¹) is an isomorphism for i < 2n - 1 and a surjection for i = 2n - 1. From this follows that πᵢ(X) → πᵢ₊₁(SX) for all (n-1)-connected CW complexes. Since, as established, πₙ(Sⁿ) ≈ ℤ for all positive n, the degree map between the two is also isomorphic. A pair of r-connected and s-connected CW complexes X, A, with non-negative r, s, and the quotient map X → X/A will induce a map πᵢ(X, A) → πᵢ(X/A) which is either isomorphic for i ≤ r + s, and surjective for i = r + s + 1.
Some spaces have only one nontrivial homotopy group. Such cases are called "Eilenberg-MacLane space", written K(G(=πₙ(X)), n). Such a CW complex is uniquely characterized, and can be constructed for arbitrary n > 1 and abelian G by beginning with an (n-1)-connected CW complex with dimension n + 1, then attaching a higher-dimensional cell to make the homotopy group trivial for i > n. The Hurewicz Theorem can follow from this: An (n-1)-connected space X with n ≥ 2, the universal cover of H(X) vanishes for i < n, and the homotopy group is isomorphic to Hₙ(X). For a pair (X, A) where A is simply-connected and nonempty, then Hₙ(X, A) = 0 for i < n. More concisely, the Hurewicz map h: πₙ(X, A, x₀) → Hₙ(X, A) is defined by h([f]) = f⁎(α), where α is a fixed generator of Hₙ(Dⁿ, ∂Dⁿ) ≈ ℤ, and f⁎ is induced by f. It's homomorphic for n > 1, and πₙ(X, A, x₀) is a group. If A is instead merely nonempty, then the Hurewicz map is isomorphic and Hₙ(X, A) = 0 for i < n.
For a sensible quotient group X / A, there should exist some inclusion of A into X. This logic also extends to their homology groups (not homotopy groups, again, because of excision). To introduce this logic to homotopy groups, one can construct a "fiber bundle", which carries more homogeneity. Take the bundle F → E → B with p: E → B. The subspaces of p's inverse are homeomorphic "fibers". p is considered to have the "homotopy lifting property" with respect to a space X, if for a homotopy gₜ: X → B and map h₀: X → E, so that ph₀ = g₀, there is a homotopy lifting gₜ the same way. p is a "fibration", if it has this homotopy lifting property with respect to all spaces X.
Assume p has the homotopy lifting property w.r.t all disks. Given basepoints in B and F (the inverse of p on the basepoint), the map p⁎: πₙ(E, F, x) → πₙ(B, b) is isomorphic for all positive n. Since B is path connected, this relation generates a long exact sequence. The map is also considered to have the homotopy lifting property for a pair (X, A), if each homotopy from X into B lifts to one from X into E, for lifts up to t (which leads from A into E). Maps satisfying this condition for disks is also referred to as "Serre fibrations". "Fibre bundles" one a space E with fiber F are maps p: E → B with neighborhoods for all points in B, so that homeomorphisms h: p⁻¹(U) → U ⨉ F, so that it commutes with U ⨉ F→ U, p: p⁻¹(U) → U. h is considered a "local trivialization" of the bundle. This structure can also be written as F → E → B, with base space B and total space E. Fibre bundles have the homotopy lifting property w.r.t all CW pairs.
Connections with Cohomology
Finally, using the concept of Eilenberg-MacLane spaces, a homotopy construction of cohomologies becomes possible. Given natural bijections T: ⟨X, K(G, n)⟩ → Hⁿ(X; G) for CW complexes X and n > , with any abelian group G. T([f]) = f*(h) some distinguished class h ∈ Hⁿ(K(G, n); G). The transformation T is supposed to be an isomorphism, so some group structure needs to exist on ⟨X, K(G, n)⟩. It will turn out that this group structure is natural. The class h as described, is considered a "fundamental class", since the space it exists in is isomorphic to Hom(Hₙ(K; ℤ), G) as per Hurewicz. For some chosen CW complex with (n-1)-skeleton point K(G, n), the fundamental class is represented by the cellular cochain assigning elements πₙ(K(G, n)) to each n-cell for K(G, n). For connected CW complexes, this also holds true for nonbasepointed homotopy classes. For a proof by construction, functors hⁿ(X) = ⟨X, K(G, n)⟩ define a reduced cohomology theory on the (category of) basepointed CW complexes, and reduced cohomology theories h* on CW complexes with coefficient groups hⁿ(S⁰) nonzero only for n = 0 implies the existence of natural isomorphisms hⁿ(X) for all CW complexes X and all n.
To supplement the lack of sum operator on ⟨X, K⟩, introduce the "reduced suspension" of X, written ΣX, which is the sum restricted to maps sending the whole segment {x₀} ⨉ I to the basepoint. For CW complexes with a 0-cell as basepoint, the quotient map SX → ΣX is a homotopy equivalence, for collapsing (contractible) subcomplexes of SX to a point. This identifies ⟨SX, K⟩ with ⟨ΣX, K⟩. While this achieves what was nominally required, it's much more interesting to gain the group structure from K rather than from X. For this, use an adjoint relation: ⟨ΣX, K⟩ = ⟨X, ΩK⟩ with ΩK the space of loops ("loopspace") in K at its basepoint. The constant loop is taken as the basepoint of ΩK. ΩK is topologized as a subspace of K⁻¹ with compact-open topology. The sum arises as composition of loops. Since cohomology groups are abelian, ⟨X, ΩK⟩ should also be abelian, which can be achieved through iterating on n-loopspaces ⟨X, ΩⁿK⟩ (= ⟨ΣⁿX, K⟩) which are abelian for all n > 1.
Sequences of CW complexes K₁, K₂, ... with weak homotopy equivalences Kₙ → ΩKₙ₊₁ is an "Ω-spectrum". For Ω-spectrum {Kₙ}, the functors X → hⁿ(X) = ⟨X, Kₙ⟩, n ∈ ℤ define a reduced cohomology theory on the basepointed CW complexes and basepoint-preserving maps. The converse is also true. Such spaces Kₙ are also considered "infinite loopspaces" since weak homotopy equivalences Kₙ → ΩᵐKₙ₊ₘ for all m. For most purposes, cohomology theories don't change with respect to their background of basepointed or nonbasepointed CW complexes. Some reduced basepointed cohomology theory r* can always be lead to an unreduced one, through hⁿ(X, A) = rⁿ(X/A). For X, assume that X/Ø = X₊ is the union of X with some disjoint basepoint. The resulting theory is nonbasepointed, since the basepoint is preserved under maps. The inverse direction uses the co-kernel of the map projecting from the basepoint into the entire space X. If h* is an unreduced cohomology theory on the category of CW pairs and hⁿ(x₀) = 0 for all n except for 0, then natural isomorphisms hⁿ(X, A) ≈ Hⁿ(X, A; hⁿ(x₀)) emerges for all (X, A) and n. The same is true for homology theories.
For fibrations p: E → B ⇒ Fₙ = p⁻¹(n) ∀ n ∈ B (n are path components of B). Given this, the fibrations might adhere to a homotopy analog to the local triviality property of fiber bundles, which are inherent to the object anyways. The restriction on a fibration p: p⁻¹(A) → A is itself a fibration for any subspace of B. Two different fibrations p, q may be connected by a "fiber-preserving" map f, if p = qf. Such maps are "fiber homotopy equivalence", if has an inverse map g, such that fg and gf are homotopic to the identity map. They can be thought of as families of homotopy equivalences between the fibers originating from the two source spaces. If a fibration happens to project into the same spaces as a map, a pullback of the for f*(E) → A exists. If there are several of these pullbacks, they are of course fiber homotopy equivalent. In the same vein, fibrations over contractible bases B are fiber homotopy equivalent to a product fibration B ⨉ F → B.
Since there is clearly some proximity between fibrations and maps, there might be a way to turn maps into fibrations. Given some map f: A → B and E the space of pairs (a, g) so that g: I → B is a path in B with g(0) = f(a) and a is some element of A. E can be topologized as a subspace of A and the space of mappings I → B with the compact-open topology. Maps p: E → B, p(a, g) = g(1) are fibrations. If A is a subspace consisting of (a, g) with g being constant paths at f(a), and if E deformation retracts into A through restriction of g to increasingly shorter initial segments, the map p: E → B restricts to f on A. This extends f to a fibration E → B. The resulting fiber is called the "homotopy fiber" of f. If f is already a fibration, then this construction creates an inclusion that is a fiber homotopy equivalence. The homotopy fibers are then homotopy equivalent to the actual fibers. If F → E → B is a fibration or fiber bundle with contractible E, then a weak homotopy equivalence F → ΩB exists. Using fibers, limits in the sense of category theory can be constructed for a chain. These are "Postnikov towers".