Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.32426

Associative Lifts of CW Semigroups

Stefano Bonzio·Andrea Loi
View original source ↗Read source document ↗Export metadata ↓

Abstract

Every nonempty connected finite-dimensional CW topological semigroup admits a continuous associative multiplication on its universal cover that lifts the multiplication of the base. Such a multiplication is unique up to a deck transformation: in fact, any two associative lifts are conjugate whenever they exist. We show that the finite-dimensional hypothesis cannot be dropped in general: a connected, countable, infinite-dimensional CW topological semigroup is constructed for which continuous lifts of multiplication exist, but none is associative. The counterexample has nonabelian fundamental group isomorphic to Thompson's group $F$. Thus finite dimensionality is sufficient for associative lifting in the CW setting, whereas the unrestricted infinite-dimensional statement fails.

Explore related subjects

math.ATmath.GNmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefano Bonzio, Andrea Loi. 2026-10-23. Associative Lifts of CW Semigroups. https://arxiv.org/abs/2610.32426

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Corrections and additions to "A note on the hit problem for the polynomial algebra in the case of odd primes and its application" [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 118 (2024), article 22]

In this work, we correct results in the published paper [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 118 (2024), article 22]. We determine the non-hit kernel of the first reduced power in polynomial degree $p+1$, identify the published Frobenius construction as zero on indecomposables, and replace the multiplier assertion by a theorem with explicit degree hypotheses. We also classify the monomial multipliers of Frobenius that preserve hits in one variable. For the third algebraic transfer, we correct the target dimensions and establish surjectivity throughout the four principal degree families. Explicit invariant calculations and Minami's degree reduction give isomorphisms in seven infinite families at $p=3$ and $p=5$, including a family with an initial change from dimension one to dimension two. For every odd prime, compatibility with the Frobenius operation on $\operatorname{Ext}$ gives an exact sequence comparing successive transfer kernels and forces a nonzero invariant extraction kernel at every second-family dimension jump. We compute the first cohomology of a nonzero extraction kernel and determine the exact obstruction to injectivity from rank-two products in the first and fourth families. Coefficient extraction and changes of coordinates give an additional infinite isomorphism family and a spanning result for annihilated polynomial homology. At each extraction isomorphism, the quotient by all rank-two products is preserved as a general linear group module. We distinguish these results from the unresolved invariant dimensions at arbitrary normalized index differences, and give the precise limitations of composition-factor counts and Dickson multiplication. An appendix supplies explicit representatives for all the nonzero rank-two invariant lines in Crossley's dimension classification.

math.AT↗
arXiv2026

Minimal finite models of wedges of spheres with at most eight points

A minimal finite model of a topological space is a finite $T_0$-space of least cardinality that is weak homotopy equivalent to it. We determine, up to homeomorphism, the minimal finite models of eight wedges of one- and two-dimensional spheres. The spaces $S^{2}\vee S^{1}$ and $S^{2}\vee S^{2}$ have exactly two and three minimal finite models, respectively, all with seven points, whereas $S^{1}\vee S^{1}\vee S^{2}$ has exactly seven minimal finite models, $S^{1}\vee S^{1}\vee S^{1}\vee S^{2}$ has one, $S^{1}\vee S^{2}\vee S^{2}$ has six, $S^{2}\vee S^{2}\vee S^{2}$ has five, $S^{2}\vee S^{2}\vee S^{2}\vee S^{2}$ has three, and $S^{1}\vee S^{1}\vee S^{2}\vee S^{2}$ has two, all with eight points. To prove this we classify the connected finite $T_0$-spaces of height $2$ without beat points that have seven or eight points, and we identify their weak homotopy types by means of McCord's theorem and explicit collapse and suspension arguments for their two-dimensional order complexes. Together with the known results for spheres and wedges of circles, this determines, up to weak homotopy equivalence, all path-connected spaces whose minimal finite models have at most eight points.

math.AT↗