Introducing H space along with the Spanier’s text Algebraic Topology.
The proof in the article is basically same in the book, except that I put somewhat more detail.
For any given category C, and object , we always have a covariant functor defined by for an object of set. And for any morphism of C, let for any , then the morphism function is defined by for any element ().
We can check the identity law: , and the composite law:
For given category C, X the object of C, is called initial object if for any object Y, consists of only one element of morphism in C. We can characterize the initial object of a category C with the functor as:
Discrete subcategory of is always regarded as a family of sets (because the morphisms are only the identity of those objects). Thus the equivalence between category in this sense does not indicate they are same; rather, this is because we can instantly observe that there is 1 to 1 correspondence if X is initial object in C.
construction of the category of directed system
is directed set. is directed system (i.e. the sequence of morphisms and objects which satisfy the following conditions).
(2) for all
The object of the category is again a sequence of morphisms s.t.
commutes. Note that the range Z is fixed. And for morphism of the category, is a map s.t.
commutes. The initial object of the category C, called direct limit of the direct system . We can construct the category of and the inverse limit of the inverse system, in a similar manner.
category of pointed topological space
We denote as the homotopy category of pointed topological space. The objects are pointed spaces denoted by (the base point is determined uniquely for each topological space). The morphisms are homotopy classes of base point preserving continuous maps between pointed topological spaces. Shortly we denote in place of (the composition law for the morphisms is confirmed based on the known fact that
“the composition of homotopic maps rel are homotopic rel if ( are homotopic maps)”.
Thus the composition can uniquely determined independent to the choice of the representation from the homotopy class).
When a topological group is given, defines a contravariant functor
Admitting the above fact without proof(c.f. Spanier pp.34), we rather want to obtain a functor without assuming that P is a group.
To do this, we shall assume that P is a pointed space, and consider the situation where has group structure but not the set of basepoint-preserving continuous maps from X to P.
Let P’ is a topological group, is an equivalence in (we don’t assume P is a group), and . Then the diagram
commutes and the vertical arrows are isomorphisms (in the category of group induced from topological group P’ by pointwise multiplication) and thus and are naturally equivalent and the output of the group structure are unchanged when in homotopy. becomes group with
condition of a functor to have a range in
We have this result.
P: pointed space. Then P is an H group iff is a contravariant functor
First we define some terminologies.
H space is a pointed topological space P together with a continuous multiplication
for which the (unique) constant map is a homotopy identity (i.e. the composition maps
are homotopic to .
homomorphism of H spaces P, P’ with multiplications and is a continuous map s.t. following diagram :
is homotopy commutative. This definition may sound instinctive when we regard the homomorphism of H spaces as continuous map compatible with the continuous multiplication of H spaces in .
We will prove the following theorem, and then the theorem 5 follows immediately.
For the reason that the techniques used in the proof seems somewhat common when it comes to put an algebraic-like structure to a topological object, which repeatedly appears in the further discussion, we give a detailed proof.
A pointed space having the same homotopy type as an H space (or an H group) is itself an H space (or H group) in such a way that the homotopy equivalence is a homomorphism.
pf. Let and be homotopy inverses and let be an H space with multiplication . Define to be the composite
is a continuous multiplication in and the composite
equals the composite
, which is homotopic to the composite . Because , the map is homotopic to . Similarly, the map is homotopic to . Therefore P’ is an H space (we checked is the homotopy identity right now). Because the square
is homotopy commutative (just compose from the right of ), is a homomorphism (by definition), and so is (since , the vertical arrow can be reversed). If is homotopy associative or homotopy abelian, so is , and if is a homotopy inverse for P, then is a homotopy inverse for P’.
We prove the last statement.
1. is associative (when is)
Consider the following diagram.
Then we can obtain
from the diagram (*1). The last homotopy equivalence can be attained with another diagram replacing with , respectively.
2. is homotopy abelian (when is)
In the similar manner above, consider the following diagram.
The composition of the bottom arrows, is homotopic to . So we get
3. is a homotopy inverse for P’ (if is for P)
Again, consider the following diagram.
I’m actually unclear about the existence of (I mean, as a non-trivial map naturally induced by the given homotopy inverse ), but if it exists, then it must commutes on the left side of diagram and it must agree with .
Given an H space P, for any pointed space X there is a law of composition in defined by
The resultant of the multiplication is a homotopy class of continuous map
and this is well-defined (again is a set of morphisms in !)
If P is a H group, then for , the inverse is . Since for any ,
Therefore, we have the theorem 5 (partially, but not discussing here although reverse is also true).