Finite Semigroup Classifying Space

• Classifying space of a finite semigroup is the geometric realization of its nerve that encodes algebraic operations into a topological framework.
• The construction generalizes group classifying spaces by incorporating unique minimal ideals and efficient algorithms for computing the associated group completion.
• Key insights include diverse homotopy types, exotic torsion phenomena, and closure under suspension and joins, highlighting distinctions from classical group theory.

The classifying space of a finite semigroup $S$, denoted $BS$, is the geometric realization of the nerve $\mathcal N_\bullet S$ defined by the combinatorial structure of $S$. This construction encodes the algebraic properties of $S$ in topological terms and generalizes the widely studied case of group classifying spaces $BG$. Recent research has established new algorithmic and structural results for $BS$, revealed a diverse range of possible homotopy types, and demonstrated phenomena exclusive to semigroups, with notable distinctions from classic group theory (Sweeney, 7 Feb 2025).

1. Fundamental Construction

Let $S$ be a finite semigroup. Consider the nerve $\mathcal N_\bullet S$ as a $\Delta$-set: $\mathcal N_n S = S^n\qquad (n \geq 1)$ with face maps

$$d_i(x_1,\ldots,x_n) = \begin{cases} (x_2, \ldots, x_n) & i=0, \ (x_1, \ldots, x_{i-1},\, x_i x_{i+1},\, x_{i+2}, \ldots, x_n) & 1 \leq i < n, \ (x_1, \ldots, x_{n-1}) & i=n. \end{cases}$$

If $S$ is a monoid, degeneracy maps are given by inserting the identity at position $i$. The classifying space is defined as the geometric realization: $BS = |\mathcal N_\bullet S|$ Alternatively, $BS$ may be modeled as a CW-complex with $n$-cells labelled by $n$-tuples in $S^n$, with attaching maps corresponding to multiplication.

2. Group Completion and the $K$-Thin Case

A central structural property of semigroups is the existence of a unique minimal two-sided ideal: $K(S) = \bigcap \{\text{nonzero ideals of } S\}.$ By the Rees–Suschkewitsch theorem, $K(S) \cong \mathcal M(H; I, J; \cdot)$ for some group $H$ and index sets $I, J$. A semigroup is “$K$-thin” if $|I| = 1$ or $|J| = 1$, i.e., $K(S)$ is left- or right-simple.

The group completion $G(S)$ is the group constructed from $S$ by universal property: any semigroup map $S \to \Gamma$ into a group factors uniquely through $S \to G(S)$. For all $S$,

$\pi_1(BS) \cong G(S).$

A principal homotopy result (“Thin–is–grouplike”) establishes that for $K$-thin finite $S$, the inclusion of the maximal subgroup $H \subset K(S)$ induces a homotopy equivalence: $BS \simeq BG(S).$ This is proven by a reduction to the inclusion of submonoids with suitable idempotents and the subsequent application of homotopy-collapsing arguments (Sweeney, 7 Feb 2025).

3. Algorithms for Computing $G(S)$

For finite $S$ presented as a multiplication table, $G(S)$ can be computed in $O(|S| \log|S| \cdot \alpha(|S|))$ operations (with $\alpha$ the inverse Ackermann function):

• Find an element $k = x_1 x_2 \cdots x_{|S|}$; then $k \in K(S)$.
• Define $I = \{xk \mid (xk)^2 = xk\}$ and $J = \{kx \mid (kx)^2 = kx\}$.
• Define $H = \{k x k \mid x \in S\}$.
• Normalize the sandwich matrix for $K(S) \cong \mathcal M(H; I,J;\cdot)$ and compute group inverses in $H$.
• Form generators for a normal subgroup $N \triangleleft H$ using $h j i h^{-1}$ for $h \in H$, $i \in I$, $j \in J$.
• Compute $N$ using union-find algorithms.
• The group completion is $G(S) = H/N$, with coset representatives used for computation.

This achieves efficient computation in practice, outperforming the naive $O(|S|^2)$ approach, making large-scale homological studies feasible.

4. Homology and Novel Phenomena

The (integral) homology $H_i(BS; \mathbb Z)$ is given by the Eilenberg–MacLane formula: $$H_i(BS; \mathbb Z) \cong \mathrm{Tor}_i^{\mathbb Z S}(\mathbb Z, \mathbb Z).$$ For finite $S$ of order up to $8$, almost all ($99.999\%$ for $|S|=8$) are $K$-thin and have homology identical to some finite group. However, non-$K$-thin cases reveal new phenomena: infinitely generated free homology, non-periodic rank growth, large exotic torsion, and Moore-space patterns—features without direct analogs in group homology. Computations leverage:

• “$K$-thin short circuit” reductions to group homology,
• Recursion and caching for ultimately periodic resolutions,
• Smith normal form for direct boundary matrix computations.

Specific examples refuting earlier conjectures of Nico (1969) include contractible $BS$ for non-zero semigroups, semigroups with trivial $\pi_1$ but infinite higher homology, and cases where $BS$ realizes Moore or suspension spaces.

Sample Table: Non-$K$-thin Homology Behavior

$S$ (order)	$H_2$	$H_3$	$H_6$
5-elt infinite $\mathbb Z$	$\mathbb Z$	$\mathbb Z$	$\mathbb Z$
6-elt doubling	$0$	$\mathbb Z^1$	$\mathbb Z^8$
9-elt Moore $M(\mathbb Z/2,3)$	$0$	$0$	$0$
10-elt exotic torsion	$\mathbb Z$	$\mathbb Z^3$	$\mathbb Z^9\times C_{1494640}$

The sample illustrates infinite generation and giant torsion in classes of small, non-$K$-thin semigroups.

5. Suspension, Joins, and Closure Properties

The classifying space construction for finite monoids displays closure under (reduced) suspension, and more generally, under topological join. For any finite monoid $S$ and finite discrete set $Y$,

$B J^Y(S) \simeq BS * Y$

where $J^Y(S)$ denotes a monoid formed by joining copies indexed by $Y$ with join-like rules. For $|Y|=2$,

$B J^{\{1,2\}}(S) \simeq \Sigma BS.$

Consequently, the set $\{BS \mid S \text{ finite monoid}\}$ is closed under suspension, mirroring the ability to realize new and diverse homotopy types outside the group context.

6. Illustrative Cases and Realizations

Explicit constructions realize a wide range of topological spaces as $BS$ for suitable $S$:

• Rectangular bands: For $\mathrm{Rect}^a_b$, $BS \simeq \bigvee^{(a-1)(b-1)} S^2$. $\mathrm{Rect}^2_2$ realizes $S^2$.
• Moore-space semigroup: A $9$-element $S$ yields $BS\simeq M(\mathbb Z/2,3)$ with no corresponding subgroup of order $2$.
• Suspension of $\mathbb R\mathrm P^2$: A $12$-element example produces $BS \simeq \Sigma(\mathbb R\mathrm P^2)$.
• Exponential growth: A $6$-element semigroup provides $H_i(BS) \cong \mathbb Z^{2^{i-2}}$ for $i \geq 2$, with unbounded rank.
• Large torsion: A $10$-element case yields $H_6(BS) \cong \mathbb Z^9 \times C_{1,494,640}$, exhibiting homological torsion unrelated to any subgroup order.

These cases exemplify the flexibility and complexity of semigroup classifying spaces, which support phenomena inaccessible to group theory.

7. Context, Significance, and Open Directions

The classifying space of a finite semigroup bridges algebraic and topological approaches, extending beyond group-theoretic analogs both structurally and computationally. Efficient algorithms for $G(S)$ and homological invariants allow broad exploration, while closure under suspension and joins opens rich territory for the realization of new homotopy types. The confirmation of counterexamples to longstanding conjectures demonstrates that semigroup topology is strictly richer, with patterns—such as exotic torsion growth and Moore-space realizations—not possible in group theory. A plausible implication is a greater diversity of homotopy types is accessible via semigroup classifying spaces than via group classifying spaces, motivating continued investigation into the interplay between semigroup algebra and topological invariants (Sweeney, 7 Feb 2025).