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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 16

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 16

Find the homology, cohomology and homotopy groups

a. \(H_*(\mathbb {R}P^3,\mathbb {Z})\);

b. \(H_*(\mathbb {R}P^3,\mathbb {Z}/2\mathbb {Z})\);

c. \(H^*(\mathbb {R}P^3,\mathbb {Z})\);

d. \(\pi _k(\mathbb {R}P^3,x_0)\), for \(k=1,2,3\).

Raw LaTeX

Find the homology, cohomology and homotopy groups

\textbf{a.} \(H_*(\mathbb{R}P^3,\mathbb{Z})\);

\textbf{b.} \(H_*(\mathbb{R}P^3,\mathbb{Z}/2\mathbb{Z})\);

\textbf{c.} \(H^*(\mathbb{R}P^3,\mathbb{Z})\);

\textbf{d.} \(\pi_k(\mathbb{R}P^3,x_0)\), for \(k=1,2,3\).

Topic Tags

Cohomology and Cup Products Homotopy Equivalence, Retracts, and Contractibility Homotopy Groups Homotopy and Degree on Manifolds