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There is a definition of $K^n$ for positive $n$, without Bott periodicity. This approach goes back to Karoubi, and you can find it in his book "K-theory". The definition (for both, positive and negati …

Your question is not so clearly stated; so I take the opportunity to interpret it and write a little essay. As far as I understand your question, you found out that the multiplicativity result by "Bo …

You missed that the sequence is not exact at $K(X,Y)$ (neither is it exact at $K(Y)$, but that does not matter here). There is an ambiguity coming from $K^{-1} (Y)$, i.e. automorphisms of bundles. If …

As you already said, if $E$ is flat of rank $r$, then $ch(E)=0$, meaning that the class $[E] \in K^0 (X)$ is torsion and therefore $m[E]=0 \in K^0(X)$ for some $m$. Pick $m$ such that $mr >> dim X$. …

You seem to misunderstand the grading a bit. Look at $X=CP^2$ and the $K$-theory class $H-1$, where $H$ is the Hopf bundle. Then $ch(H-1)= 1+z+\frac{z^2}{2}-1$, with $z$ a generator of $H^2 (CP^2;Z)$. …

The Fredholm index of an elliptic operator only depends on the symbol class. Here is the proof (which I memorize from Lawson-Michelsohn and Atiyah-Singer). If $D: \Gamma(E_0) \to \Gamma(E_1)$ has ord …

If $M$ is a homotopy sphere of dimension $4k>0$, then the signature is clearly zero. By the Hirzebruch signature theorem, you get $0=\langle L_k (TM); [M] \rangle = b_k \langle p_k (TM); [M] \rangle$ …

Whether the following is useful might depend on your concrete example. Because you are mentioning $K^1 $ instead of $K^0$, I assume that your Dirac operator is ungraded (if it is graded, the index sho …

Ad question 1: Let $E$ be any ring spectrum, with associated cohomology and homology theory. An $E$-orientation of a rank $n$ vector bundle $V \to X$ is a class $u \in E^n (V,V-0)$ such that the restr …