Problem In Algebraic Topology Research Articles

Complexity of Supersymmetric Systems and the Cohomology Problem

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA-complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k-local Cohomology problem and showing that it is QMA1-hard and, for a large class of instances, is contained in QMA. We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1, and for a large class of instances is contained in BQP. In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l-dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.

Geometric Structures in Group Theory

The conference was in the area of geometric group theory, the field of mathematics in which one studies infinite groups (finitely generated, or more generally locally compact, countable etc.) via actions on spaces endowed with various structures (geometric, measurable, analytic etc.). The surging current activity in the field is drawing more and more connections with other mathematical areas, and this was successfully reflected in the program of this week, during which problems in algebraic topology, representation theory and functional analysis, to name just a few, featured prominently alongside core topics in the area.

Eigenvalue topology of non-Hermitian band structures in two and three dimensions

In the band theory for non-Hermitian systems, the energy eigenvalues, which are complex, can exhibit non-trivial topology which is not present in Hermitian systems. In one dimension, it was recently noted theoretically and demonstrated experimentally that the eigenvalue topology is classified by the braid group. The classification of eigenvalue topology in higher dimensions, however, remained an open question. Here, we give a complete description of eigenvalue topology in two and three dimensional systems, including the gapped and gapless cases. We reduce the topological classification problem to a purely computational problem in algebraic topology. In two dimensions, the Brillouin zone torus is punctured by exceptional points, and each nontrivial loop in the punctured torus acquires a braid group invariant. These braids satisfy the constraint that the composite of the braids around the exceptional points is equal to the commutator of the braids on the fundamental cycles of the torus. In three dimensions, there are exceptional knots and links, and the classification depends on how they are embedded in the Brillouin zone three-torus. When the exceptional link is contained in a contractible ball, the classification can be expressed in terms of the knot group of the link. Our results provide a comprehensive understanding of non-Hermitian eigenvalue topology in higher dimensional systems, and should be important for the further explorations of topologically robust open quantum and classical systems.

Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${\Q}(n)$ in degrees less than or equal to $12.$

On the nonexistence of elements of Kervaire invariant one

We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. ${\Q}(n)$ has been explicitly calculated for $n=1,2,3,4$ but problems remain for $n \geq 5.$ In this note we show that if $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$ and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ are an admissible monomials, (that is, $u$ and $v$ meet a criterion to be in a certain basis for ${\Q}(k)$ and ${\Q}(r)$ respectively), then for each permutation $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible. As an application we consider a few cases when $n=5.$

Some Relations between Admissible Monomials for the Polynomial Algebra

LetP(n)=F2[x1,…,xn]be the polynomial algebra innvariablesxi, of degree one, over the fieldF2of two elements. The mod-2 Steenrod algebraAacts onP(n)according to well known rules. A major problem in algebraic topology is of determiningA+P(n), the image of the action of the positively graded part ofA. We are interested in the related problem of determining a basis for the quotient vector spaceQ(n)=P(n)/A+P(n).Q(n)has been explicitly calculated forn=1,2,3,4but problems remain forn≥5. BothP(n)=⨁d≥0Pd(n)andQ(n)are graded, wherePd(n)denotes the set of homogeneous polynomials of degreed. In this paper, we show that ifu=x1m1⋯xn-1mn-1∈Pd′(n-1)is an admissible monomial (i.e.,umeets a criterion to be in a certain basis forQ(n-1)), then, for any pair of integers (j,λ),1≤j≤n, andλ≥0, the monomialhjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1)(n)is admissible. As an application we consider a few cases whenn=5.

Using Membrane Computing for Effective Homology

Effective Homology is an algebraic-topological method based on the com- putational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important com- putability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cellofthecomplex.Inthispaper,bytakingasinputapixel-based2Dbinaryobject,we present a logarithmic-time uniform solution for describing a chain homotopy operator φ for its adjacency graph. This solution is based on Membrane Computing tech- niques applied to the spanning forest problem and it can be easily extended to higher dimensions.

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Given an ensemble of N×N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N→∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f m show a visually stunning convergence to the semi-circle as m→∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f m is the product of a Gaussian and a certain even polynomial of degree 2m−2; the formula is the same as that for the m×m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.

The Arf-Kervaire problem in algebraic topology: Sketch of the proof

We provide a sketch of the proof of the non-existence of Ker- vaire Invariant one manifolds using equivariant homotopy theory (15). A treatment of the statement and history of the problem can be found in The Arf-Kervaire problem in algebraic topology: Introduction. Our goal here is to introduce the reader to the techniques used to prove the result and to familiarize them with the kinds of computations needed.

Sheafhom: Software for Sparse Integer Matrices

Sheafhom is a free software package for large-scale computations in the category of finitely-generated modules over the integers and related rings. Its front end is a language for problems in algebraic topology and geometry. These problems come down to sparse systems of linear equations over the integers. Sheafhom's back end solves the sparse systems, with emphasis on avoiding fill-in and integer explosion. We survey and compare algorithms for integer sparse matrices, and we present implementation techniques in Common Lisp. The final section finds the quotient of the free abelian group on 26 letters by sets of words in the dictionary, in the spirit of [8].

Truncated resolution model structures

Using the dual of Bousfield–Friedlander localization, we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful for studying realization and moduli problems in algebraic topology.

The realization space of a $Pi;-algebra: a moduli problem in algebraic topology

A Π -algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R ( A ) of realizations of A , which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with π ∗ (X)=A , of the classifying space of the monoid of self homotopy equivalences of X . Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R ( A ), in which the space at the bottom is BAut ( A ) and the successive fibres are determined by Π -algebra cohomology. (This cohomology is the analog for Π -algebras of the Hochschild cohomology of an associative ring or the André-Quillen cohomology of a commutative ring.) It seems clear that the deformation theory can be applied with little change to study other moduli problems in algebra and topology.

The realization space of a Π-algebra: a moduli problem in algebraic topology

A Π-algebra A is a graded group with all of the algebraic structure possessed by the homotopy groups of a pointed connected topological space. We study the moduli space R( A) of realizations of A, which is defined to be the disjoint union, indexed by weak equivalence classes of CW-complexes X with π ∗(X)=A , of the classifying space of the monoid of self homotopy equivalences of X. Our approach amounts to a kind of homotopical deformation theory: we obtain a tower whose homotopy limit is R( A), in which the space at the bottom is BAut( A) and the successive fibres are determined by Π-algebra cohomology. (This cohomology is the analog for Π-algebras of the Hochschild cohomology of an associative ring or the André-Quillen cohomology of a commutative ring.) It seems clear that the deformation theory can be applied with little change to study other moduli problems in algebra and topology.

The Computability Problem in Algebraic Topology

The Computability Problem in Algebraic Topology

A conjecture of Barratt-Jones-Mahowald concerning framed manifolds having Kervaire invariant one

To SETTLE the question of the existence or non-existence of a framed manifold having a nontrivial Kervaire invariant (or Arf invariant) is one of the main long-standing problems in algebraic topology. The Kervaire invariant is a Z/Zvalued invariant which may be formulated in many contexts. Originally it occurred as an invariant in framed surgery theory and for this approach the reader may consult Cl23 for example. It is reformulated in [7] in terms of the Adams spectral sequence for the stable homotopy of spheres. In particular the only open cases were reduced to determining whether ht E Exts ‘*+‘(Z/2,2/2) is an infinite cycle, producing a non-trivial element 8, in the 2’[+’ 2 stem of the stable homotopy of spheres. More recently the Kahn-Priddy theorem [8] and the algebraic Kahn-Priddy theorem [lo] have been used to convert it to a problem in the stable homotopy of infinite dimensional real projective space [WP”. The Kahn-Priddy theorem gives a stable map T: [WF’aSO which is a split surjection of stable homotopy groups (localized at the prime 2)

Relation Between Higher Obstructions and Postnikov Invariants

The problem of extending a continuous map is one of the most important problems in algebraic topology. Many topologists have contributed for the solution of this problem. One of the most powerful methods in the extension problem is the obstruction theory defined first by S. Eilenberg [1] and developed by many others. N. E. Steenrod worked on the primary obstruction and showed that there is a strong connection between obstruction theory and cohomology operations [7].

Some Problems in Algebraic Topology and the Theory of Fibre Bundles

Some Problems in Algebraic Topology and the Theory of Fibre Bundles