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We establish that the natural permutation representation of highly transitive groups cannot be realized as the full group of automorphisms of an idempotent finite-dimensional evolution algebra acting on the set of lines spanned by natural elements. For instance, for any sufficiently large integer $n$, and $k\geq 4$, there exists no idempotent evolution algebra $X$ such that $\dim X=n$, and $\operatorname{Aut}(X)$ is isomorphic to a proper $k$-transitive subgroup of $S_n$. However, we demonstrate that for any (not necessarily faithful) permutation representation $\xi\colon G \to S_n$ and any field $\Bbbk$, there exists an idempotent finite-dimensional evolution $\Bbbk$-algebra $X$ such that $\operatorname{Aut}(X)\cong G$ and the induced representation given by the $\operatorname{Aut}(X)$-action on the natural idempotents of $X$ is equivalent to $\xi$., Comment: 14 pages, no figures

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection $\{M_i\}_{i=1}^n$ of finitely generated $\Z G$-modules that admit a submodule decomposition on which $G$ acts by permuting the summands. Then we prove the existence of connected finite spaces $X$ that realize each $M_i$ as its $i$-th homology, $G$ as its group of self-homotopy equivalences $\E(X)$, and the action of $G$ on each $M_i$ as the action of $\E(X)$ on $H_i(X; \Z)$., Comment: 14 pages, 3 figures, v3: first author's affiliation modified; grant for the second author added

In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given., Comment: 22 pages, 1 figure, v4: An error in the previous versions has been fixed without altering the main results; references have been updated

In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra., Comment: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is published under the Creative Commons Attribution license and can be freely dowloaded it from https://doi.org/10.1007/s13398-023-01414-w

An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allow us to prove that all, but one, of the known examples of simply-connected inflexible manifolds are not strongly inflexible., Comment: 26 pages, no figures; v2. overall improved version; v3. fully revised version

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of $\operatorname{Aut}(C)$ to $G(C)$, the set of grouplike elements of $C$. Furthermore, we show that $V$ can be regarded as a subset of $G(C)$ invariant through the $G$-action, and that the composition of the inclusion $G\hookrightarrow\operatorname{Aut}(C)$ with the restriction $\operatorname{Aut}(C)\to\operatorname{Sym}(V)$ is precisely $\rho$. We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully., Comment: 13 pages

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal C = \mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained., Comment: 24 pages. Minor corrections. To appear in Collectanea Mathematica

Let $X$ be a finite type $A_n^2$-polyhedron, $n \geq 2$. In this paper we study the quotient group $\mathcal{E}(X)/\mathcal{E}_*(X)$, where $\mathcal{E}(X)$ is the group of self-homotopy equivalences of $X$ and $\mathcal{E}_*(X)$ the subgroup of self-homotopy equivalences inducing the identity on the homology groups of $X$. We show that not every group can be realised as $\mathcal{E}(X)$ or $\mathcal{E}(X)/\mathcal{E}_*(X)$ for $X$ an $A_n^2$-polyhedron, $n\ge 3$, and specific results are obtained for $n=2$., Comment: Minor corrections, to appear in Journal of Group Theory

In this paper we construct an infinite family of homotopically rigid spaces. These examples are then used as building blocks to forge highly connected rational spaces with prescribed finite group of self-homotopy equivalences. They are also exploited to provide highly connected inflexible and strongly chiral manifolds., Comment: 19 pages (with an Appendix by Pascal Lambrechts and Don Stanley)

We raise the question of realizability of group actions which is an extended version of the 1960's Kahn realizability problem for (abstract) groups. Namely, if $M$ is a $\mathbb ZG$-module for a group $G$, we say that a simply-connected space $X$ realize this action if, for some $k$, $\pi_k(X)$ as a $\mathbb Z \mathcal E (X) $-module for the group $\mathcal E (X)$ of self-homotopy equivalences of $X$, is isomorphic to $M$ as a $\mathbb ZG$-module. Which modules can be so realized? In this paper we obtain a positive answer for any faithful finitely generated $\mathbb Q G$-module, where $G$ is finite. Our proof relies on providing a positive answer to Kahn's problem for a large class of orthogonal groups of which, by using invariant theory, our case is shown to be a particular one., Comment: 13 pages, no figures

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we compare with classical cocategory, but also with the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck's Torus Theorem, and corresponding results for $p$-compact groups and $p$-Noetherian groups., Comment: 20 pages, no figures

Apart from simply-connected spaces, a non simply-connected co-H-space is a typical example of a space X with a co-action of $B\pi_1(X)$ along $r^X : X \rightarrow B\pi_{1}(X)$ the classifying map of the universal covering. If such a space X is actually a co-H-space, then the fibrewise p-localization of $r^X$ (or the `almost' p-localization of X) is a fibrewise co-H-space (or an `almost' co-H-space, resp.) for every prime p. In this paper, we show that the converse statement is true, i.e., for a non simply-connected space X with a co-action of $B\pi_1(X)$ along $r^X$, X is a co-H-space if, for every prime p, the almost p-localization of X is an almost co-H-space., Comment: 10 pages, no figures

In this paper we prove that the inductive cocategory of a nilpotent $CW$-complex of finite type $X$, $\indcocat X$, is bounded above by an expression involving the inductive cocategory of the $p$-localisations of $X$. Our arguments can be dualised to LS category improving previous results by Cornea and Stanley. Finally, we show that the inductive cocategory is generic for 1-connected $H_0$-spaces of finite type., Comment: 9 pages, no figures

Ovarian fibromatosis (OF) is a rare non-neoplastic condition, more frequent in young females and characterised by ovarian enlargement with proliferation of collagen-producing spindle cells in the stroma. It usually presents with abdominal pain, menstrual disorders or as a solid mass. Hirsutism and virilisation are possible. The relevance of this subject remains in the usual misdiagnosis as a malignant tumour, which may lead to unnecessary oophorectomies. We report the case of a 17-year-old female who presented with acute intense pain in the right iliac fossa and vomits. On examination, she complained of intense pain without any signs of peritonism. An abdominal ultrasound was performed, finding an enlarged right adnexal containing a heterogenic cyst without flow to the ovary. A Pfannenstiel laparotomy showed a right ovarian torsion. The haemorrhagic cyst was drained and the ovary was detorsed. Bilateral ovarian biopsies were performed as both ovaries showed an indurated and cerebroid surface, suggestive of malignancy. Tumoural markers were negative. Histological examination confirmed OF. After 3 months, the magnetic resonance imaging showed an enlarged right ovary with a fibrous capsule surrounding both ovaries. In conclusion, OF can also be found incidentally in the context of an ovarian torsion. Since its appearance may be dismissed as malignant, it is important to recognise it and remain conservative. Biopsies can be taken to make the differential diagnosis., Competing Interests: None