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Abstract The Eden cell growth model is a simple discrete stochastic process which produces a “blob” (aggregation of cells) in $$\mathbb {R}^d$$ $R^d$: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers at timetgrow at a rate between$$t^{(d-1)/d}$$ $t^{(d-1)/d}$and$$P_d(t)$$ $P_d\left(t\right)$, where$$P_d(t)$$ $P_d\left(t\right)$is the size of the site perimeter. Assuming a widely believed conjecture, this establishes the rate of growth of the Betti numbers in every dimension. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes. 

Given a simplicial pair (X, A), a simplicial complex Y, and a map f:A -> Y, does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable. 

Spaces with positive weights are those whose rational homotopy type admits a large family of “rescaling” automorphisms. We show that finite complexes with positive weights have many genuine self-maps. We also fix the proofs of some previous related results. 

What is the maximum number of holes enclosed by a $$d$$-dimensional polyomino built of $$n$$ tiles? Represent this number by $$f_d(n)$$. Recent results show that $$f_2(n)/n$$ converges to $1/2$. We prove that for all $$d \geq 2$$ we have $$f_d(n)/n \to (d-1)/d$$ as $$n$$ goes to infinity. We also construct polyominoes in $$d$$-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems.