The Topological Entropy Conjecture

For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

Tilting classes over commutative rings

We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri, Pospíšil, ŠÅ¥ovíček and Trlifaj (2014). We show that the n-tilting classes can equivalently be expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: {\operatorname{Tor}_{*}(R/I,-)}, Koszul homology, Čech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of Šťovíček, Trlifaj and Herbera (2014). Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.

On Küneth’s correlation and its applications

Let {\mathcal{K}} be an abelian category that has enough injective objects, let {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let {T^{(i)}} be a right derived functor, {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If {T^{(i)}=0} for {i\geq 2} and {T^{(i)}C_{n}=0} for all {n\in\mathbb{Z}} , then there is an exact sequence 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where {C_{*}=\{C_{n}\}} is a chain complex in the category {\mathcal{K}} , {H_{n}(C_{*})} is the homology of the chain complex {C_{*}} , {TC_{*}} is a chain complex in the category A, and {H_{n}(TC_{*})} is the homology of the chain complex {TC_{*}} . This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence \displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*}) \displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , {T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})} . The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {\{U\}} of open subsets U of X such that {\overline{U}} is a compact subset of X.

The fundamental class of smooth Kuranishi atlases with trivial isotropy

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program.We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC — a cobordism class of smooth manifolds) and a virtual fundamental class (VFC — a Cech homology class). We, moreover, show that such Kuranishi atlases exist on simple Gromov–Witten moduli spaces and develop the technical results in a manner that easily transfers to more general settings.