# An introduction to intersection homology theory by Frances Kirwan, Jonathan Woolf

By Frances Kirwan, Jonathan Woolf

Now extra region of a century previous, intersection homology conception has confirmed to be a strong device within the learn of the topology of singular areas, with deep hyperlinks to many different components of arithmetic, together with combinatorics, differential equations, staff representations, and quantity thought. Like its predecessor, An creation to Intersection Homology conception, moment variation introduces the facility and sweetness of intersection homology, explaining the most principles and omitting, or purely sketching, the tough proofs. It treats either the fundamentals of the topic and a variety of functions, delivering lucid overviews of hugely technical parts that make the topic available and get ready readers for extra complex paintings within the zone. This moment version comprises fullyyt new chapters introducing the speculation of Witt areas, perverse sheaves, and the combinatorial intersection cohomology of fanatics. Intersection homology is a big and starting to be topic that touches on many facets of topology, geometry, and algebra. With its transparent reasons of the most rules, this ebook builds the arrogance had to take on extra professional, technical texts and gives a framework during which to put them.

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This holds since the kernel is a quotient of Hc2 (X \ Y , E), which is zero by the affine vanishing theorem of Artin–Grothendieck. Notice that X\Y = X\Y is affine of dimension 3. The divisor Y is singular, a union of products of two modular curves together with the components of the boundary divisor X \ X, which itself is a union of Shioda surfaces. Passing to the normalization of Y , one can control the cohomology groups Hc2 (Y, E|Y ) in terms of the cohomology groups of modular curves and canonical coefficient systems on them as in [110].

We still assume that πf is cuspidal, but not a CAP representation, and contributes to degree 3, however not exclusively. • (X, E(πf )). These two numbers occur among the four “roots” of the spinor L-series at the prime p. Since the Weyl group acts transitively on the four “roots” of the local spinor L-function Lp (πp , s), we can assume p3/2 α0 = pχ(p) without restriction of generality. Furthermore, the Weyl group allows us to interchange α0 α1 and α0 α2 . So we can assume p2 χ(p) to be either p3/2 α0 α1 or p3/2 α0 α1 α2 .

Thus, we obtain χ(Lγ , dgf in ) χ(Fν ) = vol (ξK (k)Af in ξ −1 ∩ Lγ (Af in )) dg f in ν for a constant χ(Lγ ) = χ(Lγ , dgf in ) depending only on Lγ and on the choice of the Haar measure dgf in on Lγ (Af in ). 4. Observe that γ is R-elliptic, and hence is Q-elliptic. For the ambiguity of this notion, see [44], p. 392. We show that in our situation this ambiguity does not cause problems, since the Euler characteristic of the corresponding summands in the trace formula vanishes unless both notions agree.