H space (spanier)

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.

covariant functor \pi_X
For any given category C, and object X\in C, we always have a covariant functor \pi_X:C\to {\bf Set} defined by \pi_X(Y)=hom(X,Y) for an object of set. And for any morphism of C, let f\in hom(Y,Z) for any Y,Z\in Ob(C), then the morphism function is defined by f_*(\lambda)=f\circ \lambda for any element \lambda\in hom(X,Y) (f_*=\pi_X(f):\pi_X(Y)\to \pi_X(Z)).

We can check the identity law: \pi_x(id_Y)(\lambda)=(id_Y)_*(\lambda)=\lambda, \forall \lambda\in \pi_X(Y), and the composite law:

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initial object

For given category C, X the object of C, is called initial object if for any object Y, hom(X,Y) consists of only one element of morphism in C. We can characterize the initial object of a category C with the functor \pi_X as:

    \[X:\ initial\ object\ of\ C\Leftrightarrow\ {\rm Im}\pi_X \text{is equivalent to a discrete subcategory of}\ {\bf Set}.\]

Discrete subcategory of {\bf Set} 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 \pi_X(Y)\leftrightarrow Y if X is initial object in C.

construction of the category of directed system

\Lambda is directed set. \{Y^\alpha,f_\alpha^\beta\}_\Lambda is directed system (i.e. the sequence of morphisms and objects which satisfy the following conditions).

(1) f_\alpha^\alpha=id_{Y^\alpha}
(2) f_\alpha^\gamma=f_\beta^\gamma f_\alpha^\beta for all \alpha\leq \beta\leq \gamma

The object of the category is again a sequence of morphisms \{g_\alpha : Y^\alpha \to Z\}_\lambda \in Ob(C) s.t.
the diagram

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commutes. Note that the range Z is fixed. And for morphism of the category, h\in hom(\{g_\alpha : Y^\alpha \to Z\}_\lambda,\{g'_\alpha : Y^\alpha \to Z'\}_\lambda) is a map h:Z\to Z' s.t.

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commutes. The initial object of the category C, called direct limit of the direct system \{Y^\alpha,f_\alpha^\beta\}_\Lambda. We can construct the category of inv\{Y^\alpha,f_\alpha^\beta\}_\Lambda and the inverse limit of the inverse system, in a similar manner.

category of pointed topological space

We denote {\bf hTop^*} as the homotopy category of pointed topological space. The objects are pointed spaces denoted by (X,x_0)(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 [X;P] in place of hom((X,x_0),(P,p_0)) (the composition law for the morphisms is confirmed based on the known fact that

“the composition of homotopic maps rel A,\ B are homotopic rel A if f_1(A)\subset B (f_1,f_2:(X,A)\to (Y,B) are homotopic maps)”.

Thus the composition can uniquely determined independent to the choice of the representation from the homotopy class).

When a topological group P is given, \pi^P defines a contravariant functor

    \[\pi^P : {\bf hTop}^*\to {\bf Grp}\]

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 [X;P] has group structure but not the set of basepoint-preserving continuous maps from X to P.

Let P’ is a topological group, F:P\to P' is an equivalence in {\bf hTop}^* (we don’t assume P is a group), and f\in [X;Y]. Then the diagram

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commutes and the vertical arrows are isomorphisms (in the category of group induced from topological group P’ by pointwise multiplication) and thus \pi^P and \pi^{P'} are naturally equivalent and the output of the group structure are unchanged when P\sim P' in homotopy. [X;P] becomes group with

    \[\lambda_1\lambda_2(x)=(F\circ\lambda_1)(x)(F\circ\lambda_2)(x),\ (F:P\simeq P',\ \lambda_1,\lambda_2\in [X;P])\]

condition of a functor \pi^P to have a range in {\bf Grp}

We have this result.

Theorem 5.

P: pointed space. Then P is an H group iff \pi^P is a contravariant functor

    \[\pi^P:{\bf hTop}^*\to {\bf Grp}\]

First we define some terminologies.

H space is a pointed topological space P together with a continuous multiplication

    \[\mu:P\times P\to P\]

for which the (unique) constant map c:P\to P is a homotopy identity (i.e. the composition maps

    \[P\overset{(c,1)}{\to} P\times P \overset{\mu}{\to} P,\ P\overset{(1,c)}{\to} P\times P \overset{\mu}{\to} P\]

are homotopic to 1_P.

homomorphism of H spaces P, P’ with multiplications \mu and \mu' is a continuous map \alpha:P\to P' s.t. following diagram :

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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 {\bf hTop^*}.

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.

Theorem 4.

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 f:P\to P' and g:P'\to P be homotopy inverses and let P be an H space with multiplication \mu:P\times P\to P. Define \mu':P'\times P'\to P' to be the composite

    \[P'\times P'\overset{g\times g}{\to} P\times P\overset{\mu}{\to} P \overset{f}{\to} P'\]

\mu' is a continuous multiplication in P' and the composite

    \[P'\overset{(1,c')}{\to} P'\times P' \overset{\mu'}{\to}P'\]

equals the composite

    \[P'\overset{g}{\to} P \overset{(1,c)}{\to} P\times P \overset{\mu}{\to} P\overset{f}{\to} P'\]

, which is homotopic to the composite P'\overset{g}{\to} P \overset{f}{\to} P'. Because fg\simeq 1_{P'}, the map \mu'\circ (1,c') is homotopic to 1_{P'}. Similarly, the map \mu'\circ (c',1) is homotopic to 1_{P'}. Therefore P’ is an H space (we checked c' is the homotopy identity right now). Because the square

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is homotopy commutative (just compose f\times f from the right of \mu'), g is a homomorphism (by definition), and so is f (since \mu'\circ (f\times f)=f\circ\mu\circ (g\times g\circ f\times f) \sim f\circ \mu, the vertical arrow can be reversed). If \mu is homotopy associative or homotopy abelian, so is \mu', and if \phi:P\to P is a homotopy inverse for P, then f\phi g:P'\to P' is a homotopy inverse for P’.

We prove the last statement.

1. \mu' is associative (when \mu is)

Consider the following diagram.

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Then we can obtain

    \[\mu'\circ (\mu'\times 1)\sim f\circ (\mu\circ (1\times \mu))\circ g^3 \sim \mu' \circ (1\times \mu')\]

from the diagram (*1). The last homotopy equivalence can be attained with another diagram replacing \mu\times 1 with 1\times \mu, respectively.

2. \mu' is homotopy abelian (when \mu is)

In the similar manner above, consider the following diagram.

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The composition of the bottom arrows, \mu\circ T is homotopic to \mu. So we get

    \[\mu'\circ T\sim f\circ \mu\circ (g\times g)=\mu'\]

3. f\phi g:P'\to P' is a homotopy inverse for P’ (if \phi:P\to P is for P)

Again, consider the following diagram.

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I’m actually unclear about the existence of \phi' (I mean, as a non-trivial map naturally induced by the given homotopy inverse \phi), but if it exists, then it must commutes on the left side of diagram and it must agree with f\phi g.

Given an H space P, for any pointed space X there is a law of composition in [X;P] defined by

    \[[g_1][g_2]=[\mu\circ (g_1,g_2)]\]

The resultant of the multiplication is a homotopy class of continuous map

    \[x\mapsto g_1(x)g_2(x)\]

and this is well-defined (again [X;P] is a set of morphisms in {\bf hTop}^*!)

If P is a H group, then for [g]\in [X;P], the inverse is [\phi\circ g]. Since g(x)\in P for any x\in X,
[g][\phi\circ g]=[\mu\circ (g,\phi\circ g)]=c.

Therefore, we have the theorem 5 (partially, but not discussing here although reverse is also true).

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