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If $X$ is a finite nonsimply-connected graph, show that $H^{n}\left(X ; \mathbb{Z}\left[\pi_{1} X\right]\right)$ is zero unless $n=1,$ when it is the direct sum of a countably infinite number of $\mathbb{Z}$ 's. [Use Proposition 3H.5 and compute $H_{c}^{n}(\tilde{X})$ as $\lim _{\text {roposition }} H^{n}\left(\tilde{X}, \tilde{X}-T_{i}\right)$ for a suitable sequence of finite subtrees $\left.T_{1} \subset T_{2} \subset \cdots \text { of } \tilde{X} \text { with } \cup_{i} T_{i}=\tilde{X} .\right]$

Calculus 3

Chapter 3

Cohomology

Section 11

Local Coefficients

Vectors

Oregon State University

Harvey Mudd College

Baylor University

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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