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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 18

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 18

Let \(X=\Delta -\bigcup _{j=1}^{\infty }D_j^\circ \) where \(\Delta \) is the closed unit disk and \(D_j^\circ \subset \Delta \) are open round disks with pairwise disjoint closures. Prove that the homology of \(X\) is infinitely generated.

Raw LaTeX

Let \(X=\Delta-\bigcup_{j=1}^{\infty}D_j^\circ\) where \(\Delta\) is the closed
unit disk and \(D_j^\circ\subset\Delta\) are open round disks with pairwise disjoint
closures. Prove that the homology of \(X\) is infinitely generated.

Topic Tags

Cellular Homology Homology with Coefficients Relative Homology and Long Exact Sequences