For finite nilpotent groups $J$ and $N$, suppose $J$ acts on $N$ via automorphisms. We exhibit a decomposition of the first cohomology set in terms of the first cohomologies of the Sylow $p$-subgroups of $J$ that mirrors the primary decomposition of $H^1(J,N)$ for abelian $N$. We then show that if $N \rtimes J$ acts on some non-empty set $\Omega$, where the action of $N$ is transitive and for each prime $p$ a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then $J$ fixes an element of $\Omega$.

For a partially multiplicative quandle (PMQ) ${\mathcal {Q}}$ we consider the topological monoid $\mathring {\mathrm {HM}}({\mathcal {Q}})$ of Hurwitz spaces of configurations in the plane with local monodromies in ${\mathcal {Q}}$. We compute the group completion of $\mathring {\mathrm {HM}}({\mathcal {Q}})$: it is the product of the (discrete) enveloping group ${\mathcal {G}}({\mathcal {Q}})$ with a component of the double loop space of the relative Hurwitz space $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$; here $G$ is any group giving rise, together with ${\mathcal {Q}}$, to a PMQ–group pair. Under the additional assumption that ${\mathcal {Q}}$ is finite and rationally Poincaré and that $G$ is finite, we compute the rational cohomology ring of $\mathrm {Hur}_+([0,1]^2,\partial [0,1]^2;{\mathcal {Q}},G)_{\mathbb {1}}$.

We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty $-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

In this paper, we present a smooth framework for some aspects of the “geometry of CW complexes”, in the sense of Buoncristiano, Rourke and Sanderson. We then apply these ideas to Morse theory, in order to generalize results of Franks and Iriye-Kono.

More precisely, consider a Morse function $f$ on a closed manifold $M$. We investigate the relations between the attaching maps in a $\text{CW}$ complex determined by $f$, and the moduli spaces of gradient flow lines of $f$, with respect to some Riemannian metric on $M$.