Algebraic Topology

New submissions

[ total of 14 entries: 1-14 ]
[ showing up to 2000 entries per page: fewer | more ]

New submissions for Tue, 20 Mar 18

[1]
Title: On the homotopy type of $\mathrm{Sp}(n)$ gauge groups
Subjects: Algebraic Topology (math.AT)

Let $\mathcal{G}_{k,n}$ be the gauge group of the principal $\mathrm{Sp}(n)$-bundle over $S^4$ corresponding to $k\in\mathbb{Z}\cong\pi_3(\mathrm{Sp}(n))$. We refine the result of Sutherland on the homotopy types of $\mathcal{G}_{k,n}$ and relate it with the order of a certain Samelson product in $\mathrm{Sp}(n)$. Then we classify the $p$-local homotopy types of $\mathcal{G}_{k,n}$ for $(p-1)^2+1\ge 2n$.

[2]
Title: Witt groups of abelian categories and perverse sheaves
Authors: Jörg Schürmann (Universität Münster), Jon Woolf (University of Liverpool)
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a splitting relation' which is a generalisation of isotropic reduction.
The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin.
Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.

Cross-lists for Tue, 20 Mar 18

[3]  arXiv:1803.06455 (cross-list from math.GT) [pdf, ps, other]
Title: A-infinity algebras, strand algebras, and contact categories
Comments: 83 pages, 23 figures, 6 tables
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Symplectic Geometry (math.SG)

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.

[4]  arXiv:1803.06788 (cross-list from math.CO) [pdf, other]
Title: The Cohomology for Wu Characteristics
Authors: Oliver Knill
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Algebraic Topology (math.AT)

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu characteristics w_k(G) = sum_(x_1,x_2,...,x_k) w(x_1)...w(x_k) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More general is the k-intersection number w_k(G_1,...G_k), where x_i in G_i. We define interaction cohomology H^p(G_1,...,G_k) compatible with w_k and invariant under Barycentric subdivison. It allows to distinguish spaces which simplicial cohomology can not: it can identify algebraically the Moebius strip and the cylinder for example. The cohomology satisfies the Kuenneth formula: the Poincare polynomials p_k(t) are ring homomorphisms from the strong ring to the ring of polynomials in t. The Dirac operator D=d+d^* defines the block diagonal Hodge Laplacian L=D^2 which leads to the generalized Hodge correspondence b_p(G)=dim(H^p_k(G)) = dim(ker(L_p)) and Euler-Poincare w_k(G)=sum_p (-1)^p dim(H^p_k(G)) for Wu characteristic. Also, like for traditional simplicial cohomology, isospectral Lax deformation D' = [B(D),D], with B(t)=d(t)-d^*(t)-ib(t), D(t)=d(t)+d(t)^* + b(t) can deform the exterior derivative d. The Brouwer-Lefschetz fixed point theorem generalizes to all Wu characteristics: given an endomorphism T of G, the super trace of its induced map on k'th cohomology defines a Lefschetz number L_k(T). The Brouwer index i_T,k(x_1,...,x_k) = product_j=1^k w(x_j) sign(T|x_j) attached to simplex tuple which is invariant under T leads to the formula L_k(T) = sum_T(x)=x i_T,k(x). For T=Id, the Lefschetz number L_k(Id) is equal to the k'th Wu characteristic w_k(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincare formula for Wu characteristic.

[5]  arXiv:1803.06800 (cross-list from cs.CC) [pdf, other]
Title: Computational topology and the Unique Games Conjecture
Comments: Full version of a conference paper in 34th International Symposium on Computational Geometry (SoCG 2018)
Subjects: Computational Complexity (cs.CC); Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Algebraic Topology (math.AT)

Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial's 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell, FOCS 2004; SICOMP, 2007) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade.
Our results partially settle an open question of Chen and Freedman (SODA 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over $\mathbb{Z}/2\mathbb{Z}$, and we show Unique Games-completeness over $\mathbb{Z}/k\mathbb{Z}$ for large $k$.) This equivalence comes from the fact that when the structure group $G$ of the covering space is Abelian - or more generally for principal $G$-bundles - Maximum Section of a $G$-Covering Space is the same as the well-studied problem of 1-Homology Localization.
Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.

Replacements for Tue, 20 Mar 18

[6]  arXiv:1409.6943 (replaced) [pdf, ps, other]
Title: Koszul duality between $E_n$-algebras and coalgebras in a filtered category
Authors: Takuo Matsuoka
Comments: Was previously part of arXiv:1312.2562. 35 pages. Technical improvements, more descriptive terminology
Subjects: Algebraic Topology (math.AT); K-Theory and Homology (math.KT)
[7]  arXiv:1602.07500 (replaced) [pdf, ps, other]
Title: Topological comparison theorems for Bredon motivic cohomology
Comments: Corrected indices in main theorem and a few minor changes. To appear, Transactions AMS
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)
[8]  arXiv:1707.00739 (replaced) [pdf, ps, other]
Title: The odd primary order of the commutator on low rank Lie groups
Authors: Tse Leung So
Comments: 18 pages; Accepted by Topology and its Applications
Subjects: Algebraic Topology (math.AT)
[9]  arXiv:1802.06381 (replaced) [pdf, ps, other]
Title: Inverse images of generic maps and homology groups of the Reeb spaces
Authors: Naoki Kitazawa
Comments: 6 pages, 1 figure, the author corrected some mistakes in the previous version
Subjects: Algebraic Topology (math.AT)
[10]  arXiv:1507.04915 (replaced) [pdf, other]
Title: The boundary model for the continuous cohomology of Isom$^+(\mathbb{H}^n)$
Authors: Hester Pieters
Comments: 25 pages, 1 figure, changed title, revised and extended version
Subjects: Group Theory (math.GR); Algebraic Topology (math.AT)
[11]  arXiv:1612.08485 (replaced) [pdf, ps, other]
Title: Limit theorems for random cubical homology
Subjects: Probability (math.PR); Algebraic Topology (math.AT)
[12]  arXiv:1712.07734 (replaced) [pdf, ps, other]
Title: Sheaf-Theoretic Stratification Learning