A manifold is simply connected if it has no ‘holes’: Image for post
(A) is a simply connected space (B) is not simply connected
An equivalent formulation of being simply-connected is that each loop can be continuously tightened to a point.
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(A) a loop in A can be tightened to a point (B) here a loop in B gets ‘caught’ on a hole and can’t be tightened to a point
Two manifolds are homeomorphic if you can deform one into the other and back again continously. Permissable deformations include stretching, squeezing and twisting, but not ripping, tearing and puncturing. This leads to the famous equivalence between a dougnut and a mug.
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How to deform a doughnut into a mug
In topology, we want to classify all manifolds into classes where all manifolds within a certain class are all homeomorphic to each other.
In two dimensions, it is (somewhat) easy to see that if a manifold is closed and without holes then it is equivalent to a sphere. As it turns out, this is sufficient to determine whether a 2-manifold is homeomorphic to the sphere (2-sphere).
Poincaré hypothesised at the start of the 20th century that this is true also in three dimensions, that is any closed, simply connected 3-manifold is homeomorphic to the 3-sphere.
In 2002, Grigori Perelman proved this to be true by using techniques such as Ricci flow and manifold surgery.

1. an experiment in your science class lists the materials needed for the lab. it is your job, as a lab partner, to measure out 25 ml of distilled water and 2.5 grams of magnesium. what lab measuring tools would you choose to measure each substance and how would you use each tool to get the correct amounts? be sure to describe the process you would follow step-by-step. (5 points) 2.which of the following is an si base unit for measuring mass? (2 points) ampere gram meter pound 3.which of the following is an si base unit for time? (2 points) decades hours minutes seconds 4.which of the following tools should a scientist use to measure an object in milligrams? (2 points) graduated cylinder pan balance tape measure thermometer 4.which of the following tools should a scientist use to measure an object in milligrams? (2 points) graduated cylinder pan balance tape measure thermometer. 5.a pencil beside a metric ruler. the ruler is scaled from 1 centimeter to 10 centimeters, with markings for millimeters between each number. one end of the pencil is beside the 0 on the ruler, and the pencil point is beside the 5. which of the following measurements is accurate but not precise? (2 points) 5 mm 5 cm 50 mm 50 cm 6. which of the following prefixes represents the largest value? (2 points) giga hector kilo milli 7. which of the following types of graphs is best for plotting the percentages of a whole value in a data set? (2 points) bar graph circle graph histogram line graph