We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More...
Barcelo, H., Greene, C., Jarrah, A.S., Welker, V.: Homology groups of cubical sets with connections. Appl. Categ. Struct. 29(3), 415–429 (2021) Article MathSciNet Google Scholar Barcelo, H., Laubenbacher, R.: Perspectives on A-homotopy theory and its applications. Discret. Math...
Homotopy groups were first studied systematically by Witold Hurewicz in [30] and [31]. DEFINITION 3.1.1 The n-th homotopy group of X, πn(X) is the set of homotopy classes of maps from the n-sphere Sn (the space of unit vectors in Rn+1) to X which send a fixed point in Sn (...
, the unit circle in the complex plane minus the square roots of roots of ∆ K (t), into the integers. Two relations hold: γ K + (α) −γ K − (α) = H(−∆ K + (α 2 )∆ K − (α 2 )) γ K (1) = 4λ(M) (2) where H(z) is the Heaviside functi...
Here we have chosen to parametrize the circle by [0, 2π], as is more natural if we think in terms of the phase angle. We could easily have chosen the unit interval [0,1] instead. This would perhaps harmonize better with our previous definition of paths and the definitions of homotopie...
S0is defined asBool, and the circle is the suspension ofBool. Algebraic rules Groups Structure of the source The structure of the source is roughly the following: Old code (directoryold/) The old library is still present, mainly to facilitate code transfer to the new library. Once everything...
stabilized spaces or spectra are the objects of study in stable C ∗ -homotopy theory. The stable homotopy category of C ∗ -algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We ...
topological defects mathematically, the elements of homotopy groups associated with the symmetry of the order parameter space are used to classify these defects, and the exact sequences of homotopy groups can be used to examine defect transformations associated with alterations of degeneracy space4,5,...
Here S~1 is the circle, T~2 is the torus, P~2 is the projective plane and K~2 is the Klein bottle. The symbol ST(M) denotes the tangent unit circle bundle of M with respect to any Riemannian metric of M and when M is nonorientable, M denotes the orientable double cover of M...
Homotopies(同伦)Fundamentalgroup(基本群)Higherhomotopygroups Homotopicmaps Definition Twocontinuousmapsf,g:X⇒Yarecalledhomotopicf≃giff thereisacontinuousmapF:[0,1]×X→YsuchthatF(0,•)=f andF(1,•)=g. Example Themapf:[0,1]→[0,1]:x→xandg:[0,1]→[0,1]:x→0are homoto...