Question
Poincaré duality relates these two types of groups, which can both be computed from subspaces via the Mayer-Vietoris sequence. For 10 points each:
[10h] Give these two types of groups used to assign algebraic invariants to topological spaces. These two types of groups are both formed by quotienting an operator’s kernel by its image in a complex.
ANSWER: homology groups AND cohomology groups [accept in either order]
[10m] Homology is often said to measure a complex’s deviation from this property, in which the image of one map is the kernel of the next. This term also denotes a differential form that is the exterior derivative of another.
ANSWER: exact [or word forms like exactness; accept exact sequence; accept exact differential form]
[10e] In a classic application, homology groups can be used to prove Brouwer’s result about the existence of one of these points for a continuous function on a compact, convex set. These points satisfy the equation “f of x equals x.”
ANSWER: fixed points [accept Brouwer fixed point theorem]
<Morrison, Other Science>
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