For an integer $r\ge 2$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that at most $r-1$ points of $M$ are mapped to the same point in $\mathbb R^n$. The space of $r$-immersions lies"between"the embeddings and the immersions. We calculate the connectivity of the layers in the homological Taylor tower for the space of $r$-immersions in $\mathbb R^n$ (modulo immersions), and give conditions that guarantee that the connectivity of the maps in the tower approaches infinity as one goes up the tower. We also compare the homological tower with the homotopical tower, and show that up to degree $2r-1$ there is a"Hurewicz isomorphism"between the first non-trivial homotopy groups of the layers of the two towers.

For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb {R}^n$ is defined to be the space of immersions of $M$ in $\mathbb {R}^n$ such that the preimage of every point in $\mathbb {R}^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb {R}^n$ and the $k^{\text {th}}$ power of the space of injective linear maps from $\mathbb {R}^m$ to $\mathbb {R}^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is “well-known”, but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions.

We use simplicial complexes to model weighted voting games where certain coalitions are considered unlikely or impossible. Expressions for Banzhaf and Shapley-Shubik power indices for such games in terms of the topology of simplicial complexes are provided. We calculate the indices in several examples of weighted voting games with unfeasible coalitions, including the U.S. Electoral College and the Parliament of Bosnia-Herzegovina.

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into R, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into $\mathbb R^n$, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into $\mathbb R^n$, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.

We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of r-immersions, which are immersions without r-fold self-intersections. We describe the first r layers of the tower and discuss the connectivities of the associated maps. We also prove several results about r-immersions that are of independent interest.