STRATIFIED SPACES TWIGS 1. Introduction These
... L is diffeomorphic to the Poincare homology 3-sphere. Thus L is a 3-manifold whose homology coincides with that of S 3 , yet L is not diffeomorphic to S 3 . In fact, L isn’t simply-connected; its fundamental group is isomorphic to the binary icosahedral group, which has the presentation ha, b, c | a ...
... L is diffeomorphic to the Poincare homology 3-sphere. Thus L is a 3-manifold whose homology coincides with that of S 3 , yet L is not diffeomorphic to S 3 . In fact, L isn’t simply-connected; its fundamental group is isomorphic to the binary icosahedral group, which has the presentation ha, b, c | a ...
Presentation Notes
... The conclusion follows whenever the conditions of the hypothesis are satisfied. In this theorem, the conclusion is that the square of the hypotenuse is equal to the sum of the other two sides. A theorem can also be represented in the form: If p then q, where p is the Hypothesis and q is the Conclusi ...
... The conclusion follows whenever the conditions of the hypothesis are satisfied. In this theorem, the conclusion is that the square of the hypotenuse is equal to the sum of the other two sides. A theorem can also be represented in the form: If p then q, where p is the Hypothesis and q is the Conclusi ...
Compact groups and products of the unit interval
... Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff space, then (i) implies (ii), where (i) X has B™ as a subspace; (ii) there is a continuous mapping of X onto l m . Juhasz, ([8], theorem 3-18), shows that condition (ii) is equivalent to various other conditions. ...
... Remark 3 3 . We have shown in the proof of Theorem 3 1 that if X is a compact Hausdorff space, then (i) implies (ii), where (i) X has B™ as a subspace; (ii) there is a continuous mapping of X onto l m . Juhasz, ([8], theorem 3-18), shows that condition (ii) is equivalent to various other conditions. ...
1.3 Using Reasoning to Find a Counterexample
... Chapter 1- Inductive and Deductive Reasoning 1.3 Using Reasoning to Find a Counterexample Today’s Goal: To find examples that contradict given conjectures. Let’s define Counterexample: In order to disprove or make a conjecture false we must find one example of the statement that does not satisfy the ...
... Chapter 1- Inductive and Deductive Reasoning 1.3 Using Reasoning to Find a Counterexample Today’s Goal: To find examples that contradict given conjectures. Let’s define Counterexample: In order to disprove or make a conjecture false we must find one example of the statement that does not satisfy the ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.