Manuscript IX-Free@1 · IX-Free.tex · section · 33b20e96

Actual stages and the domain diagram

Contract, inherited inputs, and domains › Actual stages and the domain diagram · lines 445–511

IX-Free@1#eq:diagram

Let

T = C_{c}^{\infty} (M_{+}; C), B = span {B_{f}}, W = span {W_{g}},

and set

L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},

R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}} .

Define

L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2}, V_{Σ}^{[3]} = P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}) .

The direct sums are by actual homogeneous field degree, four and two. The underlying spaces $L_{4}, R_{4}$ coincide, but their ordered roles differ.

All presentation tensor products are algebraic. Put

E_{L} = T \otimes T, E_{R} = T \otimes T

and define

p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},

p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .

The factorization diagram is

ker v \otimes T ⟶ E \otimes T ↓ v \otimes B im (v) \otimes B λ_{j} ⇢ P_{Σ} ↑ ℓ_{j} P_{Σ},

where $v = p_{L}$ or $q_{L}$ . There is a reversed diagram on the right. The dashed arrow exists exactly when the labelled prescription annihilates the displayed relation space.

Overlap conditions occur inside $P_{Σ} \otimes P_{Σ}$ , on

D_{L} = L \otimes B, D_{R} = B \otimes R, D_{old} = P_{mc} \otimes P_{mc} .

No presentation map is assumed to be a topological quotient.

Displayed formulas in this passage

IX-Free@1#eq:diagram · line 488
$ker v \otimes T ⟶ E \otimes T ↓ v \otimes B im (v) \otimes B λ_{j} ⇢ P_{Σ} ↑ ℓ_{j} P_{Σ},$
unlabeled · line 448
$T = C_{c}^{\infty} (M_{+}; C), B = span {B_{f}}, W = span {W_{g}},$
unlabeled · line 454
$L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},$
unlabeled · line 458
$R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}} .$
unlabeled · line 463
$L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2}, V_{Σ}^{[3]} = P_{Σ} \otimes_{C} C [ℏ] / (ℏ^{4}) .$
unlabeled · line 473
$E_{L} = T \otimes T, E_{R} = T \otimes T$
unlabeled · line 478
$p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},$
unlabeled · line 482
$p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .$
unlabeled · line 504
$D_{L} = L \otimes B, D_{R} = B \otimes R, D_{old} = P_{mc} \otimes P_{mc} .$