Please email any contributions to Jonathan Woolf. Thank you!
(Proof of Lemma 7.1.5), sent by Greg Friedman, Texas Christian University
There is a gap in the proof of Lemma 7.1.5 regarding the softness of the sheaf of locally-finite intersection chains. The proof is correct for the sheaf of ordinary singular chains, but does not carry over to the sheaf of intersection chains as stated. The problem is as follows.
We begin with a chain \(Z\) in \(IS(U)\), where \(U\) is a neighbourhood of the compact set K and look at a finite covering of \(K\) by open neighborhoods \(V(x)\) of points in \(K\). The neighbourhoods are chosen so that each interesects only a finite number of simplices of \(Z\). We then construct the chain that is the sum of the simplices that intersect these neighbourhoods (with appropriate coefficients). This can be regarded as a chain in \(X\) but it may not be allowable as an intersection chain. By choosing only some of the simplices of the original chain \(Z\), and not all of them, new boundaries may be introduced, and these new boundaries may destroy the allowability of the new chain. In other words, if \(C=D+E\) is an intersection cycle in which the boundaries of \(D\) and \(E\) are non-empty, the non-allowability of the simplices in the boundary of \(D\) may render D non-allowable. So even though pulling out some simplices of \(Z\) that are sufficiently close to \(K\) and forming them into a new chain works for ordinary homology, it may not be permissable for intersection homology.
This objection is actually noted in Remark 7.1.6 regarding the confusion earlier in the literature surrounding the simplicial intersection chain sheaf! This was cleared up by Habegger in "Intersection cohomology" (A. Borel et al, 1984), but the correction requires an extra subdivision step to ensure that the chains remain allowable.
Another approach to fixing this problem can be found in "Singular chain intersection homology for traditional and super-perversities" (Greg Friedman, math.GT/0407301, to appear in Transactions of the AMS) where it is shown that the sheaf of intersection chains is homotopically fine, which implies that its hypercohomology is the homology of the global sections.
(Example 6.4.1), sent by Jakub Löwit, Institute of Science and Technology, Austria
The matrix description of the real symplectic group \(SP_{2n}(\mathbb{R})\) in this example is incorrect. The symplectic group is \(2n \times 2n\) square matrices with a block decomposition
\[
X=
\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\]
satisfying \( A^tC=C^tA\), \(B^tD = D^tB\) and \(A^tD-C^tB=1\), and not \(A=D\), \(C=-B\) and \(\det(X)\neq 0\) as written in the book. (The latter is the description of a maximal compact subgroup.)
(Long exact sequences on page 123), sent by Jakub Löwit, Institute of Science and Technology, Austria
In the two long exact sequences \(\Psi_f\) should be \(\Phi_f\), so they should read
\[
\cdots \to H^i(X_0) \to H^i(X_t) \to H^i(X_0;\Phi_f\,\mathbb{F}_X) \to H^{i+1}(X_0) \to \cdots
\]
and
\(
\cdots \to H_{i+1}(X_0) \to H^i(X_0;\Phi_f\,\mathbb{F}_X)^\vee \to H_i(X_t) \to H_{i}(X_0) \to \cdots
\)
respectively.