Fundamental Group of n-sphere for n ≥ 2

Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17].This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).Riccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyKorniłowicz Artur - Institute of Informatics, University of Białystok, Sosnowa 64, 15-887 Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449-454, 1997.Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251-260, 2004.Artur Korniłowicz. The fundamental group of convex subspaces of EnT. Formalized Mathematics, 12(3):295-299, 2004.Artur Korniłowicz. On the isomorphism of fundamental groups. Formalized Mathematics, 12(3):391-396, 2004.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in EnT. Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261-268, 2004.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Marco Riccardi. The definition of topological manifolds. Formalized Mathematics, 19(1):41-44, 2011, doi: 10.2478/v10037-011-0007-4.Marco Riccardi. Planes and spheres as topological manifolds. Stereographic projection. Formalized Mathematics, 20(1):41-45, 2012, doi: 10.2478/v10037-012-0006-0.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Duality Symmetries for N=2 Supersymmetric QCD with Vanishing beta-Functions

We construct the duality groups for N=2 Supersymmetric QCD with gauge group SU(2n+1) and N_f=4n+2 hypermultiplets in the fundamental representation. The groups are generated by two elements $S$ and $T$ that satisfy a relation $(STS^{-1}T)^{2n+1}=1$. Thus, the groups are not subgroups of $SL(2,Z)$. We also construct automorphic functions that map the fundamental region of the group generated by $T$ and $STS$ to the Riemann sphere. These automorphic functions faithfully represent the duality symmetry in the Seiberg-Witten curve.Comment: 20 pages, 3 figures, harvmac (b); v2, typos corrected, statement about curves of marginal stability is correcte

Embedding spheres in knot traces

The trace of the n-framed surgery on a knot in S3 is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each n, this provides conditions that imply a knot is topologically n-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice

Embedding spheres in knot traces

The trace of $n$-framed surgery on a knot in $S^3$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice

The Alexander polynomial of (1,1)-knots

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot, which we call the n-cyclic polynomial. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S^2\times S^1, a result obtained by J. Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some properties of the Alexander polynomial of knots in the 3-sphere are extended to the case of (1,1)-knots in lens spaces.Comment: 11 pages, 1 figure. A corollary has been extended, and a new example added. Accepted for publication on J. Knot Theory Ramification

On the Yamabe Problem concerning the compact locally conformally flat manifolds

AbstractFor all known locally conformally flat compact Riemannian manifolds (Mn, g) (n > 2), with infinite fundamental group, we give the complete proof of Aubin's conjecture on scalar curvature. That solves the Yamabe Problem for these manifolds. There exists a metric g′ conformal to g, such that volg′ = 1 and whose scalar curvature R′ is constant and satisfies R′ < n(n − 1) ωn2n, where ωn is the volume of the sphere Sn with radius 1