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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 8

Denote by \([X,Y]\) the set of free homotopy classes of maps \(X\to Y\). Show that \([S^2,\mathbb {R}P^2]\) is infinite. In other words, construct a sequence of maps \(f_i:S^2\to \mathbb {R}P^2\), \(i=1,2,3,\ldots \), such that no two are homotopic to each other.

Raw LaTeX

Denote by \([X,Y]\) the set of free homotopy classes of maps \(X\to Y\).
Show that \([S^2,\mathbb{R}P^2]\) is infinite. In other words, construct a sequence of
maps \(f_i:S^2\to\mathbb{R}P^2\), \(i=1,2,3,\ldots\), such that no two are homotopic to
each other.

Topic Tags

Chain Complexes and Chain Homotopy Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Real Projective Spaces