Timeline for Local index formula for >ungraded< elliptic operators

6 events

when toggle format	what		by	license	comment
Dec 8, 2014 at 8:54	vote	accept	AlexE
Dec 8, 2014 at 8:54	answer	added	AlexE		timeline score: 0
Dec 3, 2014 at 8:27	comment	added	AlexE		Yes, I assume that $P$ is essentially self-adjoint so that it indeed defines an element of $K_1(M)$. I also assume the manifold to be compact (and without boundary). Applying the homological Chern character (constructed by using Connes' cyclic cohomology), we get an element of $H_\ast(M)$ and I want to identify the Poincare dual of it. If $P$ is graded the Poincare dual is, up to some universal constants, $ind(P)$ which is computed / constructed using the clutching construction. This is a generalization of Atiyah-Singer. But what is this Poincare dual now in the ungraded case?
Dec 2, 2014 at 21:45	comment	added	Johannes Ebert		Typically, you would replace the operator $P$ by $P \oplus P^{\ast}: E \oplus F \to E \oplus F$. This is graded, and the index (in the graded sense) is the usual index. An ungraded operator has an index in $K^1$, but only if it is self-adjoint.
Dec 2, 2014 at 16:25	comment	added	Paul Siegel		There are a number of index theorems for operators whose K-homology classes live in the odd degree groups for whatever reason, but typically you need the operator to have some extra structure. Do you have a specific operator in mind?
Dec 2, 2014 at 16:17	history	asked	AlexE	CC BY-SA 3.0