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I have a question about an argument that occurred in the discussion about consequences of Bott periodicity in A Concise Course in Algebraic Topology by P. May on page 207. Here is the excerpt:

Could anybody explain the "little argument with $H$-spaces" which May has in mind to show that $Omega^2(BU times mathbbZ) $ is equivalent to $(Omega^2_0BU) times mathbbZ$ as $H$-spaces?

My considerations: As explained above the loop space "sees" only the component of the base point so $Omega^2BU= Omega^2_0BU$.

The problem reduces to two questions:

Does $Omega^2$ respect products like $pi_k(-)$?

And which role does the fact $pi_2(BU) = mathbbZ$ play? This is a statement about homotopy classes of loops $Omega^2(BU)$ but in the considerations above we haven't passed to homotopy classes so here we would "lose" some information.

Does anybody see the correct argument? Thanks in advance.

The loop functor $Omega$ is defined on a space $X$ as $Omega X=Map_*(S^1,X)$. For compactly generated spaces you combine the adjunction homeomorphism $Map_*(S^1,Map_*(S^1,X))cong Map(S^1wedge S^1,X)$ with $S^1wedge S^1cong S^2$ and iterate to get a homeomorphism

$$Omega^kXcong Map_*(S^k,X).$$

In fact many people would take this as the definition of $Omega^k$, and use the previous isomorphisms backwards to identify

$$Omega(Omega^k-1)congOmega^k.$$

Anyway, the point is that for a space $X$ you literally have

$$pi_k(X)congpi_0(Omega^kX)$$

for each $kgeq 0$. Moreover if $Y$ is another compactly generated space then you also have

$$Omega^k(Xtimes Y)=Map_*(S^k,X^ktimes Y)cong Map_*(S^k,X)times Map_*(S^k,Y)=Omega^k XtimesOmega^k Y$$

and this follows from the properties of the mapping functor. May discusses these adjunctions in Chapter 5 Compactly Generated Spaces.

Now the statement you are interested in is actually something more general. What May is referencing is the the fact that

If $X$ is any grouplike H-space with a homotopy associative multiplication, then there is a homotopy equivalence $$Xsimeq X_0timespi_0X$$ where $X_0$ denotes the path component of $X$ containing the basepoint. Moreover, this equivalence is one of $H$-spaces if $X$ is homotopy commutative.

I'll discuss this general case below but you can apply it to your case as follows. I'll shortly give a definition for "grouplike", and you will see that any loop space is grouplike. Since a loop space is also homotopy associative - and a double loop space is homotopy commutative - we can apply the above to $Omega^2(BUtimesmathbbZ)$ to solve your problem.

Start by observing that $pi_2(BUtimesmathbbZ)congpi_2(BU)opluspi_2(mathbbZ)congpi_2(BU)congmathbbZ$ since $pi_2BU_1congpi_2K(mathbbZ,2)congmathbbZ$ and $BU_1rightarrow BU$ is 3-connected. This means that

$$pi_0(Omega^2(BUtimesmathbbZ))congpi_2(BUtimesmathbbZ)congmathbbZ.$$

The basepoint in $Omega^2(BUtimesmathbbZ)$ is the constant loop $S^2rightarrow BUtimesmathbbZ$, $tmapsto (ast,0)$ which is contained in $Omega^2_0BUcong Omega^2_0(BU)times0subseteqOmega^2(BU)timesOmega^2(mathbbZ)congOmega^2(BUtimesmathbbZ)$. Hence we have from the above that

$$Omega^2(BUtimesmathbbZ)simeq Omega^2_0(BUtimesmathbbZ)timespi_0(Omega^2(BUtimesmathbbZ))congOmega^2_0BUtimesmathbbZ$$

as H-spaces, as claimed by May.

Now, onto the discussion. An H-space $(X,m)$ is said to be grouplike if $pi_0X$ becomes a group under the operation induced by the multiplication $m$. Clearly this holds when $Xsimeq Omega X'$ is a loop space, since in this case $pi_0Xcongpi_0(Omega X')congpi_1X'$ is a group.

To see that the operation induced by $m$ coincides with the loop addition you must use the fact that $m$ extends the fold map $nabla:Xvee Xrightarrow X$, and the comutiplication $c:S^1rightarrow S^1vee S^1$, which induces the loop sum, lifts the diagonal $S^1rightarrow S^1times S^1$. If you do not see this immediately I urge you to draw a diagram, taking loops $k,l:S^1rightarrow X$ and forming the coposite $mcirc(ktimes l)circDelta$ on the top row, and the composite $nablacirc(kvee l)circ c$ on the bottom.

Now to prove the theorem, let us take a homotopy associative, grouplike H-space $X$ with multiplication $m$. Choose a representative $x_g$ for each coset in $pi_0$, making sure to pick the H-space unit $ast$ for the basepoint component. Now form the map

$$Psi:X_0timespi_0Xrightarrow X$$

by setting

$$Psi(x,[x_g])=m(x,x_g):=xcdot x_g.$$

We claim that this map is a homotopy equivalence. Indeed, it has an inverse $Theta:Xrightarrow X_0timespi_0X$ which we define as follows. Assume $xin X$ lies in $[x_g]inpi_0X$. Then since $pi_0X$ is a group the inverse $[x_g]^-1$ exists and we have $[x_g]^-1=[x_h]$ for some $x_hin X$. Set

$$Theta(x)=(xcdot x_h,x_g).$$

Clearly $[xcdot x_h]=[x_gcdot x_h]=[x_g]cdot[x_h]=[ast]$, so $xcdot x_h$ indeed lies in $X_0$ and we take the above as the definition of $Theta$.

We have

$$PsicircTheta(x)=Psi(xcdot x_h,x_g)=(xcdot x_h)cdot x_gsimeq xcdot(x_gcdot x_h)$$

and since $[x_h]=[xcdot x_g]^-1=[x_g]^-1$ we have $[x_gcdot x_h]=[ast]$ and get a homotopy $psicircThetasimeq id$ by choosing a path from $x_gcdot x_h$ to the identity for each index. Similarly we find that $ThetacircPsisimeq id$.

If in addition we assume that the multiplication on $X$ is homotopy commutative then we find

$$Psi((x,[x_g])cdot(y,[x_h]))=Psi(xcdot y,[x_g][x_h])=Psi(xcdot y,[x_gx_h])=(xcdot y))cdot x_k$$

where $[x_gcdot x_h]=[x_k]$. Continuing on we get

$$(xcdot y)cdot x_ksimeq (xcdot y)cdot (x_gcdot x_h)simeq(xcdot (ycdot x_g))cdot x_hsimeq (xcdot (x_gcdot y))cdot x_hsimeq(xcdot x_g)cdot (ycdot y_h)$$

where we have used the canonical homotopy associativity and commutativity of $m$. But this last expression is exactly $Psi(x,[x_g])cdot Psi(y,[x_h])$. Hence we have

$$Psi(-cdot-)simeq Psi(-)cdotPsi(-)$$

showing that $Psi$ is an equivalence of H-spaces. I'll leave you to check the last couple of details.