Simply Connected Moore Spaces

• Simply connected Moore spaces are defined as the atomic retract T^(2n+1){p^r} in the loop space of mod p^r Moore spaces, central to understanding torsion phenomena.
• The atomic decomposition isolates the bottom cell from ωP^(2n+1)(p^r), allowing precise analysis of p^(r+1)-power maps and strengthening classical exponent results.
• Combinatorial approaches via Cohen groups clarify how iterated commutators and power maps determine the null homotopy of the projection, deepening our grasp of modular homotopy structures.

A simply connected Moore space refers, in this context, to the atomic retract $T^{2n+1}\{p^r\}$ within the loop space of a $(2n+1)$-dimensional mod $p^r$ Moore space $P^{2n+1}(p^r)$. This space arises as a distinguished summand in a deep homotopical decomposition, playing a key role in homotopy theory, especially concerning exponent phenomena and the structure of torsion in the loop spaces of Moore spaces. For odd primes $p > 3$, $n > 1$, and $r > 1$, the atomic piece $T^{2n+1}\{p^r\}$ encapsulates the bottom cell's behavior under iterated power maps, exhibiting a refined combinatorial structure that advances understanding of classical exponent results in homotopy theory (Cohen et al., 2015).

1. Construction and Definition of Moore Spaces

For an odd prime $p$ and integer $r \geq 1$, the classical $(2n+1)$-dimensional mod $p^r$ Moore space $P^{2n+1}(p^r)$ is the homotopy cofiber of the degree-$p^r$ map on $S^{2n}$, with the cofibration sequence: $$S^{2n} \xrightarrow{p^r} S^{2n} \longrightarrow P^{2n+1}(p^r) \longrightarrow S^{2n+1}$$ Since $\pi_1(P^{2n+1}(p^r)) = 0$ for $n \geq 1$, the Moore space is simply connected in this range.

A functorial homotopy decomposition of the loop-suspension $\Omega P^{2n+1}(p^r)$ was established for $p > 2$ and $n > 1$: $$\Omega P^{2n+1}(p^r) \simeq T^{2n+1}\{p^r\} \times \Omega P(n,p^r)$$ Here, $T^{2n+1}\{p^r\}$ is the atomic retract containing the $(2n+1)$-dimensional bottom cell, while $P(n,p^r)$ denotes a wedge of higher-dimensional mod $p^r$ Moore spaces. Because $\Omega P^{2n+1}(p^r)$ is $(2n-1)$-connected for $n > 1$, $T^{2n+1}\{p^r\}$ inherits simple connectivity for $n > 1$.

2. Atomic Decomposition and Homotopy Splitting

This decomposition clarifies the homotopy structure of $\Omega P^{2n+1}(p^r)$ and isolates the atomic factor $T^{2n+1}\{p^r\}$. The retract corresponds to the unique bottom cell and shows functoriality in the splitting, with the projection $$\alpha\colon \Omega P^{2n+1}(p^r) \to T^{2n+1}\{p^r\}$$ specifying the factorization. The complementary factor $\Omega P(n,p^r)$ absorbs the remaining higher-cell torsion phenomena.

The fibre sequence defining the atomic retract after looping the classical fibration may be summarized as: $$\Omega P(n,p^r)\longrightarrow \Omega P^{2n+1}(p^r)\xrightarrow{\alpha} T^{2n+1}\{p^r\}$$

3. Main Theorem on Power Maps and Null Homotopy

The principal result for $n > 1$, $p > 3$, $r > 1$ asserts that the composite map

$$\Omega P^{2n+1}(p^r)\xrightarrow{\,p^{r+1}\,}\Omega P^{2n+1}(p^r)\xrightarrow{\,\alpha\,}T^{2n+1}\{p^r\}$$

is null homotopic; symbolically, $\alpha \circ p^{r+1} \simeq *$. Equivalently, the projection of the $p^{r+1}$-th power map of the loop space to its atomic piece vanishes. This theorem strengthens the classical result that $\Omega P^{2n+1}(p^r)$ has exponent $p^{r+1}$, by localizing the vanishing to the atomic retrace containing the bottom cell.

4. Combinatorial Approach via Cohen Groups

The proof employs a combinatorial group-theoretic framework rooted in the James construction and Cohen groups. Consider the Cohen group $K_n(p^r)$ constructed from the free group $F(x_1,\ldots,x_n)$ by imposing Milnor-type relations (left-normalized commutators with repeated indices are trivial) and $x_i^{p^r}=1$. The group $K_n(p^r)$ is nilpotent of class $n$.

Within $K_n(p^r)$, the "equalizer" subgroup $H_n(p^r)$ is the intersection $\bigcap_{i=1}^n \ker(d_i)$, where $d_i$ deletes $x_i$ and renumbers coordinates. The "fat-diagonal" element $\alpha_n = x_1x_2\cdots x_n \in H_n(p^r)$ encodes the inclusion $J_n(X)\hookrightarrow\Omega\Sigma X$ for $X=P^{2n}(p^r)$.

The $p^{r+1}$-power map on the loop space is detected by $\alpha_n^{p^{r+1}}$. Combinatorial calculus using left-normalized commutators establishes that

$\alpha_n^{p^{r+1}} \in B_n(p^r)$

where $B_n(p^r)$ is a subgroup generated by commutators of length differing from powers of $p$. $B_n(p^r)$ corresponds to self-maps factoring through the higher-cell summands, and all elements of $B_n(p^r)$ vanish under projection $\alpha$ to the atomic factor.

A key identity, illustrating the combinatorial structure, expresses iterated commutators with Stirling-type multiplicities: $$[\;x_{n+1},_{\,l}(x_1x_2\ldots x_n)\;] \;=\;\prod_{i=1}^l\;\prod_{\sigma\in E_{n,i}}\; [x_{n+1},x_{\sigma(1)},\ldots,x_{\sigma(i)}]^{d_i(\sigma)}$$ where $E_{n,i}$ enumerates breaking $\{1,\ldots,n\}$ into $i$ rising blocks, and $d_i(\sigma)$ gives the relevant multiplicity. Divisibility properties of these sums yield $p$-divisibility results forcing $\alpha_n^{p^{r+1}}$ into $B_n$, and thus its projection is null homotopic under $\alpha$.

5. Refinement Over Classical Exponent Results

Neisendorfer's classical theorem establishes that $\Omega P^{2n+1}(p^r)$ has exponent $p^{r+1}$: the $p^{r+1}$-power map is null in all homotopy groups. The atomic result sharpens this, proving the vanishing already at the level of the atomic summand $T^{2n+1}\{p^r\}$. Thus all $p^{r+1}$-torsion occurs in the complementary higher-cell factor, while the atomic piece satisfies the exponent constraint in this more refined sense.

Homotopically, this result reveals a splitting of torsion and demonstrates that towers of Moore spaces or constructions like Anick's fibration assign the full classical exponent to the bottom-cell atomic factors. Algebraically, the combinatorial approach elucidates the obstruction carried by commutators of length divisible by $p$ in the Cohen group.

6. Essential Diagrams and Structural Summary

The following diagrams and decompositions encapsulate the essential structure:

Atomic Decomposition	Fibre Sequence	Null-Homotopy Statement	Combinatorial Formula
$$\Omega P^{2n+1}(p^r) \simeq T^{2n+1}\{p^r\} \times \Omega P(n,p^r)$$	$$\Omega P(n,p^r) \to \Omega P^{2n+1}(p^r)\xrightarrow{\alpha} T^{2n+1}\{p^r\}$$	$$\alpha \circ p^{r+1}: \Omega P^{2n+1}(p^r)\to T^{2n+1}\{p^r\} \simeq *$$	$$[x_{n+1},_l(x_1\ldots x_n)] = \prod_{i=1}^l\prod_{\sigma\in E_{n,i}} [x_{n+1},x_{\sigma(1)},\ldots,x_{\sigma(i)}]^{d_i(\sigma)}$$

Each stage identifies how the $p^{r+1}$-power of the fat-diagonal element maps into high-length commutators, geometrically translates into the higher-cell complement, and hence delivers the atomic null-homotopy via combinatorial arguments.

7. Broader Significance and Future Considerations

These combinatorial insights, established by Cohen–Mikhailov–Wu (Cohen et al., 2015), suggest avenues for further refinement of exponent phenomena in loop spaces, especially in the context of localized homotopy decompositions and torsion analysis. The techniques hinge on group-theoretic calculations and James filtration, which plausibly underpin broader investigations into homotopical splittings and modular phenomena in atomic retracts of loop spaces.