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Neil Strickland
Neil Strickland
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Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles ifon $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly homotopy invariant, and the spaces $U_{\pm}$ are contractible, so $K^0(U_{\pm})=K^0(\text{point})=\mathbb{Z}$.

For a noncompact manifold $M$ we can also consider $\widetilde{K}^0(M\cup\{\infty\})$. This is the Grothendieck group of vector bundles on $M$ with a specified trivialisation outside of a compact set. This is also interesting, but different.

With the homotopical version of $K$-theory, you get a Mayer-Vietoris sequence for $K^0(A\cup B)$ whenever the map from the homotopy pushout of $A\xleftarrow{}A\cap B\to B$ to $A\cup B$ is a homotopy equivalence. Provided that all the relevant spaces have the homotopy type of CW complexes, this essentially means that you get a Mayer-Vietoris sequence in $K$-theory iff you get one in ordinary singular cohomology. In particular, this will certainly work if $A$ and $B$ are open subsets of a manifold.

Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles if $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly homotopy invariant, and the spaces $U_{\pm}$ are contractible, so $K^0(U_{\pm})=K^0(\text{point})=\mathbb{Z}$.

For a noncompact manifold $M$ we can also consider $\widetilde{K}^0(M\cup\{\infty\})$. This is the Grothendieck group of vector bundles on $M$ with a specified trivialisation outside of a compact set. This is also interesting, but different.

With the homotopical version of $K$-theory, you get a Mayer-Vietoris sequence for $K^0(A\cup B)$ whenever the map from the homotopy pushout of $A\xleftarrow{}A\cap B\to B$ to $A\cup B$ is a homotopy equivalence. Provided that all the relevant spaces have the homotopy type of CW complexes, this essentially means that you get a Mayer-Vietoris sequence in $K$-theory iff you get one in ordinary singular cohomology. In particular, this will certainly work if $A$ and $B$ are open subsets of a manifold.

Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles on $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly homotopy invariant, and the spaces $U_{\pm}$ are contractible, so $K^0(U_{\pm})=K^0(\text{point})=\mathbb{Z}$.

For a noncompact manifold $M$ we can also consider $\widetilde{K}^0(M\cup\{\infty\})$. This is the Grothendieck group of vector bundles on $M$ with a specified trivialisation outside of a compact set. This is also interesting, but different.

With the homotopical version of $K$-theory, you get a Mayer-Vietoris sequence for $K^0(A\cup B)$ whenever the map from the homotopy pushout of $A\xleftarrow{}A\cap B\to B$ to $A\cup B$ is a homotopy equivalence. Provided that all the relevant spaces have the homotopy type of CW complexes, this essentially means that you get a Mayer-Vietoris sequence in $K$-theory iff you get one in ordinary singular cohomology. In particular, this will certainly work if $A$ and $B$ are open subsets of a manifold.

Source Link
Neil Strickland
Neil Strickland
  • 56.9k
  • 7
  • 142
  • 262

Freed, Hopkins and Teleman will be using the homotopical definition $K^0(X)=[X,\mathbb{Z}\times BU]$. For many spaces $X$ this is the same as the Grothendieck group of vector bundles if $X$; in particular this holds if $X$ is compact Hausdorff, or if it is a finite-dimensional CW complex. This definition is visibly homotopy invariant, and the spaces $U_{\pm}$ are contractible, so $K^0(U_{\pm})=K^0(\text{point})=\mathbb{Z}$.

For a noncompact manifold $M$ we can also consider $\widetilde{K}^0(M\cup\{\infty\})$. This is the Grothendieck group of vector bundles on $M$ with a specified trivialisation outside of a compact set. This is also interesting, but different.

With the homotopical version of $K$-theory, you get a Mayer-Vietoris sequence for $K^0(A\cup B)$ whenever the map from the homotopy pushout of $A\xleftarrow{}A\cap B\to B$ to $A\cup B$ is a homotopy equivalence. Provided that all the relevant spaces have the homotopy type of CW complexes, this essentially means that you get a Mayer-Vietoris sequence in $K$-theory iff you get one in ordinary singular cohomology. In particular, this will certainly work if $A$ and $B$ are open subsets of a manifold.