We calculate the tropical Dolbeault cohomology for the analytifications of P1 and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincaré duality and behaves ...analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology.
Let $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to ...H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.
Let (M,g) be an open, oriented and incomplete riemannian manifold. The aim of this paper is to study the following two sequences of L2-cohomology groups:1.H2,m→Mi(M,g) defined as the image ...(H2,mini(M,g)→H2,maxi(M,g))2.H¯2,m→Mi(M,g) defined as the image (H¯2,mini(M,g)→H¯2,maxi(M,g)). We show, under suitable hypothesis, that the first sequence is the cohomology of a Hilbert complex which contains the minimal one and is contained in the maximal one. In particular this leads us to prove a Hodge theorem for these groups. We also show that when the second sequence is finite dimensional then Poincaré duality holds and that, with the same assumptions, when dim(M)=4n then we can employ H¯2,m→M2n(M,g) in order to define an L2-signature on M. We prove several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension ΔiF of Δi.
In this paper we study M(X), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map Ψ from M(X) into the quotient of K(X)=X,BSO by the action of the group ...of homotopy classes of simple self equivalences of X. The map Ψ describes which bundles over X can occur as normal bundles of manifolds in M(X). We determine the image of Ψ when X belongs to a certain class of homology spheres. In particular, we find conditions on elements of K(X) that guarantee they are pullbacks of normal bundles of manifolds in M(X).
We answer a weaker version of the classification problem for the homotopy types of (n — 2)-connected closed orientable (2n — 1)-manifolds. Let n ≥ 6 be an even integer and let X be an (n — ...2)-connected finite orientable Poincaré (2n — 1)-complex such that Hn-1 (X;ℚ) = 0 and Hn-1 (X;ℤ2) = 0. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on Hn-1 (X; ℤp) for each odd prime p. A stronger result is obtained when localized at odd primes.
This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show ...that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P/Qk and 1/Q, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple F12 hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model.
Four-Dimensional Complexes with Fundamental Class Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
Mediterranean journal of mathematics,
12/2020, Letnik:
17, Številka:
6
Journal Article
Recenzirano
This paper continues the study of 4-dimensional complexes from our previous work Cavicchioli et al. (Homol Homotopy Appl 18(2):267–281, 2016; Mediterr J Math 15(2):61, 2018.
...https://doi.org/10.1007/s00009-018-1102-3
) on the computation of Poincaré duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal
PD
4
-complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincaré duality in all dimensions. Such complexes with partial Poincaré duality properties, which we call
SFC
4
-complexes, are very interesting to study and can be classified, up to homotopy type. For this, we introduce the concept of resolution, which allows us to state a condition for a
SFC
4
-complex to be a
PD
4
-complex. Finally, we obtain a partial classification of
SFC
4
-complexes. A future goal will be a classification in terms of algebraic
SFC
4
-complexes similar to the very satisfactory classification result of
PD
4
-complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).
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