6
$\begingroup$

I am primarily a student of physics and am trying to self-learn some algebraic topology. I am having some difficulty understanding the differences between the constructions of

$(X,A)$ (Pair of spaces), $X/A$ (Quotient space), $G/H$ (Quotient group of topological groups), $G/H$ (Orbit space where H is viewed as acting on $G$ say by left multiplication)

My questions are as follows:

  1. If $G$ is a topological group and $H$ is a (normal) subgroup then is the quotient group $G/H$ (topologically) the same as $G/H$ viewed as a quotient topological space? If not is there a condition on the topologies or spaces in which they coincide? How does the orbit space $G/H$ differ from these two notions?

  2. I think I always took for granted that $(X,A)$ was the same as $X/A$ (quotient space) due to excision in homology but now that I am learning some homotopy theory I am not so sure. Is $(X,A)$ ever the same as $X/A$?

  3. Under what conditions is $\pi_n(X,A) \cong \pi_n(X/A)$

  • 0
    For 1, it would depend on the action of H, usually there is more than one way to define an action, and its left actions by multiplication that will make the space of orbits and the quotient group the same (think of orbits as cosets).2011-10-15

1 Answers 1