By Christian Bär
Providing a scientific creation to differential characters as brought through Cheeger and Simons, this article describes vital suggestions corresponding to fiber integration, greater dimensional holonomy, transgression, and the product constitution in a geometrical demeanour. Differential characters shape a version of what's these days referred to as differential cohomology, that's the mathematical constitution at the back of the better gauge theories in physics.
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There is a unique ﬁber integration of diﬀerential characters satisfying the axioms in Deﬁnition 38. Remark 46. The proof of Theorem 44 shows more than compatibility with topological trivializations of ﬂat characters. Namely, (63) commutes for all 48 Christian B¨ ar and Christian Becker η ∈ Ω k−1 (E), not necessarily closed. In other words, we have shown compatibility with topological trivializations of characters, not necessarily ﬂat. Proposition 47 (Compatibility of Fiber Integration with Characteristic Class).
H ∗ (X; Z), +, ∗) is a ring. 32 Christian B¨ ar and Christian Becker 2. Graded commutativity. e. for h ∈ H k (X; Z) and f ∈ H l (X; Z), we have f ∗ h = (−1)kl h ∗ f . 3. Naturality. For any smooth map g : Y → X and h, f ∈ H ∗ (X; Z), we have g ∗ (h ∗ f ) = g ∗ h ∗ g ∗ f . 4. Compatibility with curvature. e. for h, f ∈ H ∗ (X; Z), we have curv(h ∗ f ) = curv(h) ∧ curv(f ). 5. Compatibility with characteristic class. e. for h, f ∈ H ∗ (X; Z), we have c(h ∗ f ) = c(h) ∪ c(f ). 6. Compatibility with topological trivialization.
Lemma 43. Let k > dim F . , [z − ∂a (z)]∂Sk−dim F = [ζ (z)]k−1−dim F is true for all z ∈ Zk−1−dim F (X; Z). , ˆ (π! h)(z) := h([PBE ζ (z)]∂Sk ) · exp 2πi curv(h) a (z) F holds for all z ∈ Zk−1−dim F (X; Z) and all h ∈ H k (E; Z). Proof. Let z ∈ Zk−1−dim F (X; Z) be a cycle. Then we ﬁnd a geometric boundary ∂β(z) ∈ Bk−1−dim F (X) such that ζ (z) − ζ(z) = ∂β(z). Since [∂a(z) − ∂a (z)]∂Sk−dim F = [∂β(z)]∂Sk−dim F = ∂[β(z)]Sk−dim F , we ﬁnd a smooth singular cycle w(z) ∈ Zk−dim F (X; Z) such that [a(z) − a (z) − w(z)]Sk−dim F = [β(z)]Sk−dim F .