Remark: Poicare Duality for closed, oriented manifolds states that for an n-dimensional
manifold (where n is the dimension over R) the pairing
Hi (X, C) Hni (X ; C) C
Corollary: dimC Hi (X ; C) = dimC Hni (X ; C)
Example: Let X = cfw_x0
Fix a commutative ring R and a topological space X . An R-local system
on X is a sheaf F of R-modules such that for all x X , there exists an
open neighborhood Ux of x and R-module M such that F|Ux = M Ux - the
constant sheaf corresponding
1. Complexes of Sheaves
Denition 1.1. A complex of sheaves (dierential graded sheaf DGS)
A on X is a collection of sheaves Ai , i Z, and morphisms (dierentials) di : Ai Ai+1 so that di+1 di = 0.
Denition 1.2. The i-th cohomology sheaf of A is
Hi (A ) := K
1. Poincare Duality for pseudomanifolds
In this section we will show a generalized version of Poincar Duality. We show
Theorem 1. If X is an oriented, topological pseudomanifold of dimension n, and p
and q are complementary pervesities (i.e. p(k ) + q
Denition 1.1. A pseudomanifold X n is normal if it has connected links,
that is, x, x Ux open in X such that Ux \ Xn2 is connected.
Denition 1.2. A normalization of a pseudomanifold X n is a normal pseu
domanifold X n together with a nite
Math 853 - Review of (co)homology
September 24, 2012
We begin with some basic denitions needed to develop simplicial homology.
Denition. An n-simplex Rn is the convex hull of n + 1 points (vertices) v0 , ., vn ,
Topics in Algebraic Topology
Let X be an n-dimensional PL pseudomanifold. So:
(i) X is ltered by closed PL subsets satisfying X = Xn Xn1 = Xn2
(ii) X Xn2 is dense in X
(iii) X has locally conical structure (in the normal direction of str
MATH 853 NOTES WEEK 8
1. Towards Poincare Duality for Intersection Homology
Let X n be an n dimensional oriented pseudomanifold and p, q be complementary perversities (i.e.
p(k ) + q (k ) = k 2). Then there exists a bilinear non-degenerate pairing
Delignes Construction of Intersection Homology
and Generalized Poincar Duality
Let X be a pseudomanifold of dimension n with ltration
X = Xn Xn1 . . . X0 .
The intersection complex IC on X is a complex of sheaves such that the sheaf
ICi is dened by U
Nearby & Vanishing Cycles
Let X be a complex algebraic (or analytic) manifold, and f : X C a regular (or proper
holomorphic function). If 0 C is a critical value, then set X0 := f 1 (0) for the special
(singular) ber, and Xs := f 1 (s) for the generic (
Generalization of the Riemann-Hurwitz formula
If X is a nite CW complexes and : Y X is a d-fold covering, then
(Y ) = d (X ).
Recall that if X is a Riemann surfaces (closed oriented surface, hence a connected sum of tori), then (X ) = 2 2g
More vanishing results
Let A be a eld.
Proposition 1.1. a) If X is an orientable real topological manifold of dimension n, and
G a constructable sheaf on X , then
H k (X, G ) = 0, if k > n.
b) If X is a complex algebraic (analytic) variety of complex di
Theorem 1.1. Let X be a complex algebraic variety of pure dimension n. Assume X is a
rational homology manifold. Then ICX is quasi-isomorphic to QX [n].
Proof. Let X be a Whitney stratication of X with strata Sc indexed by complex codimension, i
Part 1: Topological invariance of intersection homology
Let X n be a pseudomanifold with a xed stratication X and p be a xed perversity. Recall
that the Deligne complex ICp,X is characterized, uniquely up to quasi-isomorphism, by the
set of axioms [AX ]p,
Part 1. Foreword
A space X is a rational homology manifold (or Q-manifold) of real
dimension n if, x X , the local homology groups at x are computed
Hi (X, X x; Q) =
Q, i = n
0, i = n
It is easy to see that locally contractible spaces (e.g., pseudomani