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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 7

Let \(n_1\), \(n_2\) and \(n_3\) be integers. Given an example of a topological space \(X\) with \[ \pi _1(X,x_0)=\langle a,b\mid a^{n_1}b^{n_2}a^{n_3}b\rangle . \] What is \(H_1(X)\)? Under what conditions is \(H_1(X)\) torsion free?

Raw LaTeX

Let \(n_1\), \(n_2\) and \(n_3\) be integers. Given an example of a topological
space \(X\) with
\[
  \pi_1(X,x_0)=\langle a,b\mid a^{n_1}b^{n_2}a^{n_3}b\rangle.
\]
What is \(H_1(X)\)? Under what conditions is \(H_1(X)\) torsion free?

Topic Tags

Fundamental Groups Homotopy Groups Point-Set Topology and Separation Theorems Smooth Manifolds and Embedded Submanifolds