Research proof index

The set that we are taking the infimum over in the geometry bound is a finite set.

Let $H$ be a subgroup of $G \times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\phi: G \to G'$ such that $H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$ In particular, $|H| = |H_0| |H_1|$.

If one grid cube $I$ is strictly contained below another grid cube $J$, then the project’s phase distance at the finer cube is controlled by the phase distance at the coarser cube:

$d_I(f,g)\le C(a)\,d_J(f,g).$

A compact Hausdorff vector space over 𝔽_p is totally disconnected.

Given map f: M ⟶ N and q : ℕ, if H^{q+1}(M) ⟶ H^{q+1}(N) is surjective, then any z : Z^{q+1}(N) can be written as f(z') + d(y) for some z' : Z^{q+1}(M) and y : C^q(M). Note that d is spelled as toCocycles.

Assume that we have a short exact sequence 0 → A → B → C → 0 in Rep R G and that the sequence of H- invariants is also a short exact in Rep R (G ⧸ H) : 0 → Aᴴ → Bᴴ → Cᴴ → 0. Then we have a commuting square Hⁿ(G ⧸ H, Cᴴ) ⟶ H^{n+1}(G ⧸ H, Aᴴ) | | ↓ ↓ Hⁿ(G , C) ⟶ H^{n+1}(G,A) where the horizontal maps are connecting homomorphisms an...