uconf.algebraic.spherical

Rank-1 sphere cochain models and explicit Surjection actions.

This module implements the reduced cochain complex of S^d as a one-dimensional module and equips it with the explicit Surjection-algebra structure.

Background. By the Berger–Fresse theorem [BF04], the normalized cochain complex C^*(Delta^N; k) is a module over the Surjection operad. For the sphere S^d, the reduced cochain complex N^*(S^d) is the one-dimensional module with generator in degree d.

A surjection u in S(n) can act non-trivially on N^*(S^d) only if its degree equals d(n-1) (so that the total input degree dn matches the output degree d). Such surjections are sphere-admissible: they decompose as the concatenation of (d+1) permutations of {1,ldots,n} with consecutive overlap of one entry (see _extract_concatenated_permutations()).

The sign of the action mu_u(g, ldots, g) = varepsilon(u) cdot g is read off from the unique non-zero AW contribution when applying the BF chain action to the top simplex (0, ldots, d) (see _sphere_surjection_basis_sign()).

Reference: C. Berger, B. Fresse, “Combinatorial operad actions on cochains”, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135–174.

Classes

ReducedSphereCochains(d, base_ring)	Reduced cochains of S^d as a rank-1 graded module.
SurjectionSphereCochainAlgebra(d, base_ring)	Explicit Surjection-algebra structure on reduced cochains of S^d.

class uconf.algebraic.spherical.ReducedSphereCochains(d, base_ring)

Bases: CombinatorialFreeModule

Reduced cochains of S^d as a rank-1 graded module.

The reduced cochain complex N^*(S^d; k) is the one-dimensional k-module spanned by the fundamental class g in degree d (using the homological grading convention deg(g) = d). The boundary is zero.

Parameters:

d (int)

sphere_dim()

Return the sphere dimension d.

Return type:

int

generator()

Return the generator of the reduced cochains.

graded_basis

/home/runner/work/uconf/uconf/src/uconf/algebraic/spherical.py (starting at line 218)

Type:

File

graded_weighted_basis

/home/runner/work/uconf/uconf/src/uconf/algebraic/spherical.py (starting at line 226)

Type:

File

ReducedSphereCochains.Element

alias of Element

class uconf.algebraic.spherical.SurjectionSphereCochainAlgebra(d, base_ring)

Bases: OperadAlgebra

Explicit Surjection-algebra structure on reduced cochains of S^d.

For a surjection u in S(n) of degree d(n-1) and the generator g in N^*(S^d), the algebra structure is

\[\mu_u(g, \ldots, g) = \varepsilon(u) \cdot g\]

where varepsilon(u) in {-1, 0, +1} is computed by _sphere_surjection_basis_sign(). For surjections of any other degree, the action is zero (degree reasons).

This is the explicit sphere model of the Berger–Fresse algebra structure [BF04], obtained by applying the general cochain action of surjection_cochain_action() to the top cochain of Delta^d and restricting to the sphere quotient.

Parameters:

d (int)