Definition & Example Quizzes

Quizzes will be announced in advance in class.

Quiz given	Terms you should know:	Be able to give an example of:
Jan. 28	Covering space (in the sense of Massey), locally path-connected, lift (of a map), covering homomorphism (also isomorphism, automorphism), semilocal simple connectivity, universal cover, regular cover, group action	A space which is PC but not LPC. A space which is PC and LPC but not SLSC. A covering map which is not the identity, and not the universal cover of the circle.
Feb. 20	n-simplex, barycentric coordinates, face (of an n-simplex), boundary (of an n-simplex), interior (of an n-simplex), ∆-complex structure, n-chain, boundary map, chain complex, n^th homology group, n-cycle, boundary (in a chain complex), homology class, homologous elements	A ∆-complex structure on a genus 2 surface.
Mar. 7	singular n-simplex, singular n-chain, singular homology group, reduced homology, chain map, chain homotopy, exact sequence, short exact sequence, homotopy of maps, deformation retract, homotopy equivalence	A space X with H0(X) = Z^k for any given k = 1, 2, 3, … An example of two spaces which are homotopy equivalent such that neither is a deformation retract of the other.
Mar. 28	Relative homology group, relative cycle, relative boundary, SES of chain complexes, barycenter, barycentric subdivision	A chain in a space X which is not a cycle in Cn(X) but which is a cycle in Cn(X,A) for some choice of subspace A in X. (Be specific!) A decomposition of the torus into subspaces A and B satisfying the hypotheses of the Mayer-Vietoris Theorem, where A and B are both connected and A is homotopy equivalent to a wedge of two circles.
Apr. 23	Cochain group, coboundary map, cohomology group, free resolution, cocycle, coboundary (check back later, I may add a few more)	A space whose first homology group is Z/pZ for any given p. A short exact sequence which is not exact when dualized. A chain complex which has at least one cohomology group which is not equal to the dual of the corresponding homology group. Make sure you can state precisely which cohomology group of your chain complex has this property! A free resolution of the abelian group Z x Z/3Z.