Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 15

True or false: Suppose that \(X\) is a finite CW-complex and \(A\subset X\) is a subcomplex. Then the relative homotopy groups \(\pi _j(X,A)\) are isomorphic to the homotopy groups \(\pi _j(X/A)\), where \(X/A\) is the CW-complex obtained from \(X\) by collapsing \(A\) to a point. Justify your answer.

Raw LaTeX

True or false: Suppose that \(X\) is a finite CW-complex and
\(A\subset X\) is a subcomplex. Then the relative homotopy groups \(\pi_j(X,A)\) are
isomorphic to the homotopy groups \(\pi_j(X/A)\), where \(X/A\) is the CW-complex obtained
from \(X\) by collapsing \(A\) to a point. Justify your answer.

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Homotopy Groups Homotopy and Degree on Manifolds Simplicial and Delta Complexes