Poincare Duality Research Articles

From gravity to string topology

The chain gravity properad introduced in Merkulov (Gravity prop and moduli spaces $${\mathcal {M}}_{g,n}$$ , 2021, http://arxiv.org/abs/2108.10644 ) acts on the cyclic Hochschild complex of any cyclic $$A_\infty $$ algebra equipped with a scalar product of degree $$-d$$ . In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree d, and that action factors through a quotient dg properad $${\mathcal{S}\mathcal{T}}_{3-d}$$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $$\bar{H}_\cdot ^{S^1}(LM)$$ of the loop space of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ comes equipped with a morphism from the gravity properad $${\mathcal {G}} rav_{3-d}$$ which is fully determined by the compactly supported cohomology of the moduli spaces $${\mathcal {M}}_{g,n}$$ of stable algebraic curves of genus g with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ is also a properad under the properad of involutive Lie bialgebras $${\mathcal {L}}{ ieb }^{\diamond }_{3-d}$$ whose induced (via $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ ) action on $$\bar{H}_\cdot ^{S^1}(LM)$$ agrees precisely with the famous purely geometric construction of Chas and Sullivan (String topology, ; in: The legacy of Niels Henrik Abel, Springer, Berlin 2004), (ii) $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ is a properad under the properad of homotopy involutive Lie bialgebras $${\mathcal {H}}{ olieb }^{\diamond }_{2-d}$$ which controls (via $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ ) four universal string topology operations introduced in Merkulov (Propof ribbon hypergraphs and strongly homotopy involutive Lie bialgebras, 2020, http://arxiv.org/abs/1812.04913 ), (iii) E. Getzler’s gravity operad injects into $$H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$$ implying a purely algebraic counterpart of the geometric construction of Westerland (Math Ann 340:97–142, 2008) establishing an action of the gravity operad on $$\bar{H}_\cdot ^{S^1}(LM)$$ .

Thurston’s fragmentation and c-principles

Abstract In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston’s technique compared to the better-known Segal-McDuff’s proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston’s fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]). To the memory of John Mather.

Intersection homology duality and pairings: singular, PL and sheaf-theoretic

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky-MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector, and Saralegi preserves product structures.

K-theory and the singularity category of quotient singularities

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ of a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and that $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincare duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups.

Isoperimetric inequalities for Poincaré duality groups

We show that every oriented n-dimensional Poincare duality group over a ∗-ring R is amenable or satisfies a linear homological isoperimetric inequality in dimension n−1. As an application, we prove the Tits alternative for such groups when n=2. We then deduce a new proof of the fact that when n=2 and R=Z then the group in question is a surface group.

Four-Dimensional Complexes with Fundamental Class

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 Poincare duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal $${\text {PD}}_4$$ -complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincare duality in all dimensions. Such complexes with partial Poincare duality properties, which we call $${\text {SFC}}_4$$ -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 $${\text {SFC}}_4$$ -complex to be a $${\text {PD}}_4$$ -complex. Finally, we obtain a partial classification of $${\text {SFC}}_4$$ -complexes. A future goal will be a classification in terms of algebraic $${\text {SFC}}_4$$ -complexes similar to the very satisfactory classification result of $${\text {PD}}_4$$ -complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).

On groups with $S^2$ Bowditch boundary

We prove that a relatively hyperbolic pair (G,P) has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.

The Principles of protein surgery and its application to the logical drug design for the treatment of neurodegenerative diseases.

Here we describe the principles of protein surgery and its application to the molecular design for regulating the protein conformation. We initially describe the Poincare duality that defines the basis of complementarity in the time-dependent geometrical space. Next we introduce the theory of protein surgery consisting of "dissection" and "suture," which correspond to differentiation and integration in a phase space, respectively. Then we introduce the surgical (pan-manifold) differential equation and solve several simple cases. In this way, we constructed "medical quantization" strategy which makes a bridge between quantum mechanics and molecular biology. As the application of this theory, here we describe a logical drug design methodology in detail, and its application to anti-prion drug design. Finally, we propose a plan for logical drug (and medical device) design center, in which all the necessary procedures could be done in one place.

On the rational homotopy type of intersection spaces

Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincare duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to Sullivan to define a suitable commutative cochain algebra model for intersection spaces. Our construction parallels Banagl's commutative cochain algebra of smooth differential forms modeling intersection space cohomology, and we show that both algebras are weakly equivalent.

Dimensions of multi-fan duality algebras

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. The algebras constructed this way include many important examples: cohomology algebras of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced and studied by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of nonsingular points are irrelevant. By using bistellar moves we show that the number of distinct dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial on general pseudomanifolds.

Alternative to Morse–Novikov theory for a closed 1-form. I

This paper extends the Alternative to Morse–Novikov theory we have proposed in Burghelea (New topological invariants for real- and angle-valued maps, Word Scientific, Hackensack, 2018) from real- and angle-valued maps to closed 1-forms. For a topological closed 1-form $$\omega $$ on a compact ANR ( $$=$$ absolute neighborhood retract) X, a concept generalizing closed differential 1-form on a compact manifold, under the mild hypothesis of tameness, a field $$\kappa $$ and a non-negative integer r, we propose two configurations $$\varvec{\delta }^\omega _r{:}\,\mathbb {R}\rightarrow \mathbb {Z}_{\geqslant 0}$$ and which recover Novikov–Betti numbers and the Novikov complex associated with a Morse closed 1-form with non-degenerated zeros. Precisely, the sum of the multiplicities of the points in the support of $$\varvec{\delta }^\omega _r$$ equals the rth Novikov–Betti number and that of the points in the support of $$\varvec{\gamma }^\omega _r$$ equals the rank of the boundary map in the Novikov complex. We formulate the basic properties of these configurations, the stability and the Poincare duality when X is a $$\kappa $$ -orientable closed topological manifold, which in full generality will be proven in the second part of this work.

Trimming a Gorenstein ideal

Let $Q$ be a regular local ring of dimension $3$. We show how to trim a Gorenstein ideal in $Q$ to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra $P$ padded with a nonzero graded vector space on which $P_{\ge 1}$ acts trivially. We explicitly construct an infinite family of such rings.

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,\omega)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,\omega) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].

On Duality in Filtered Cohomologies

For a symplectic manifold (M2n, ω) without boundary (not necessarily compact), we prove Poincare type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality between (d + dΛ)-and ddΛ-cohomologies.

The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincare duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric L-spectrum of $${\mathbb {Z}}$$ , which is, up to weak equivalence, an $$E_\infty $$ ring map. Using this map, we construct a fundamental L-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces whose fundamental group satisfies the Novikov conjecture.

Poincaré/Koszul Duality

We prove a duality for factorization homology which generalizes both usual Poincare duality for manifolds and Koszul duality for \({\mathcal{E}_n}\)-algebras. The duality has application to the Hochschild homology of associative algebras and enveloping algebras of Lie algebras. We interpret our result at the level of topological quantum field theory.

Spectral analysis of Morse-Smale gradient flows

On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of currents. In particular, we prove that this dynamical spectrum is given by linear combinations with integer coefficients of the Lyapunov exponents at the critical points of the Morse function. Via this spectral analysis and in analogy with Hodge-de Rham theory, we give an interpretation of the Morse complex as the image of the de Rham complex under the spectral projector on the kernel of the generator of the flow. This allows us to recover classical results from differential topology such as the Morse inequalities and Poincare duality.

Tropical Hodge numbers of non-archimedean curves

We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincare duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields with residue field ℂ, the tropical (1, 1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincare duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.

Poincaré duality and Langlands duality for extended affine Weyl groups

In this paper we construct an equivariant Poincare duality between dual tori equipped with finite group actions. We use this to demonstrate that Langlands duality induces a rational isomorphism between the group C ∗ -algebras of extended affine Weyl groups at the level of K -theory.

On Four-Dimensional Poincaré Duality Cobordism Groups

This paper continues the study of four-dimensional Poincare duality cobordism theory from our previous work Cavicchioli et al. (Homol. Homotopy Appl. 18(2):267–281, 2016). Let P be an oriented finite Poincare duality complex of dimension 4. Then, we calculate the Poincare duality cobordism group $$\Omega _{4}^{{\text {PD}}}(P)$$ . The main result states the existence of the exact sequence $$0 \rightarrow L_4 (\pi _1 (P))/A_4 (H_2 (B\pi _1 (P), L_2)) \rightarrow {{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P) \rightarrow \mathbb Z_8 \rightarrow 0$$ , where $${{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P)$$ is the kernel of the canonical map $${\Omega }_{4}^{\mathrm{PD}}(P) \rightarrow H_4 (P, \mathbb Z) \cong \mathbb Z$$ and $$A_4 : H_4 (B\pi _1, \mathbb L) \rightarrow L_4 (\pi _1 (P))$$ is the assembly map. It turns out that $${\Omega }_{4}^{\mathrm{PD}}(P)$$ depends only on $$\pi _1 (P)$$ and the assembly map $$A_4$$ . This does not hold in higher dimensions. Then, we discuss several examples. The cases in which the canonical map $$\Omega _{4}^{{\text {TOP}}}(P) \rightarrow \Omega _{4}^{{\text {PD}}}(P)$$ is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In fact, we establish an exact sequence where s is Ranicki’s total surgery obtruction map. In the above cases, there are $${\text {PD}}_4$$ -complexes X which cannot be homotopy equivalent to manifolds.