 
Summary: EQUIVARIANT COHOMOLOGY IN ALGEBRAIC
GEOMETRY
APPENDIX A: ALGEBRAIC TOPOLOGY
WILLIAM FULTON
NOTES BY DAVE ANDERSON
In this appendix, we collect some basic facts from algebraic topology
pertaining to the fundamental class of an algebraic variety, and Gysin push
forward maps in cohomology. Much of this material can be found in [Ful97,
Appendix B], and we often refer there for proofs.
This appendix is in rough form, and will probably change significantly.
(Watch the version date.)
1. A brief review of singular (co)homology
Let X be any space, let CX be the complex of singular chains on X, and
let CX = Hom(CX, Z) be the complex of singular cochains. The singular
homology modules are defined as
HiX = hi(CX),
and the singular cohomology modules are
Hi
X = hi
(C
