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I am reading Hatcher's Topology and in it, it is noted that a space is simply connected, by definition, if and only if it is path connected and has trivial fundamental group.

Can someone provide some insight behind the motive for studying these spaces? What is the motive behind the name "Simply Connected"?

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    In multivariable calculus, we know that integrating a line integral is path independent as long as we can "continuously deform" the path into the other without hitting any singularities. Concretely, consider the integral of $1/(x^2+y^2)$ on the path around the circle. Normally a closed loop integral would just be 0, but in this case the singularity at (0,0) prevents us from "shrinking" the path down to a point (which would make the integral 0). So path independence of line integrals is directly dependent on whether the contour is defined on a simple connected space. – Supersingularity Jun 28 '15 at 06:13
  • The integrals of closed $1$-forms on closed paths do not change when paths are deformed, that's what @Supersingularity: means. And how about https://en.wikipedia.org/wiki/Riemann_mapping_theorem ? – orangeskid Jun 28 '15 at 07:50

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Simple connection is important in different areas of mathematics:

$real$ $analysis:$ if you have an exact $1$-form $\omega$ and you need to integrate it along a closed path $\gamma$ contained in a simple connected space then

$$\int_\gamma \omega =0$$

because $\gamma$ is homotopic to a point and the integral does not change over any path in the homotopy class of $\gamma$. Moreover by the statement above; if $\gamma_1,\gamma_2$ are two different paths starting and ending at same points, then

$$\int_{\gamma_1} \omega=\int_{\gamma_2}\omega.$$

$Covering$ $space:$ In algebraic topology every ''good enough'' topological space, I mean semi-locally simply connected, locally path connected admits a simply connected covering space.

$Riemann$ $Surfaces:$ If this beautiful area of math one of the most important theorem is the Uniformization Theorem: Every simply connected Riemann Surface is biholomorphic to only one of these three domains: the sphere, the complex plane or the upper half plane.

Complex Analysis: See the Monodromy on this page. https://en.wikipedia.org/wiki/Monodromy Simple connection is the crucial hyphotesis.

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