Sketches of solutions to selected exercises
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چکیده
Let M be the closed Mobius strip (i.e. including its boundary). Let ∂M be the boundary of M . Compute H∗(M,∂M) using arguments with the long exact sequence and other basic properties of homology. You can use that you already know homology computations for circles, but you shouldn’t compute H∗(M,∂M) directly from the Delta complex. If necessary, you can also use that H∗(X) = H ∗ (X) if X can be realized as a Delta complex in order to compute H∗(M) and H∗(∂M). To get at H∗(M,∂M), we first compute H∗(M) and H∗(∂M). The boundary ∂A is homeomorphic to a circle, so, using what we know about circles (or spheres in general), H1(∂M) = Z, and H0(∂M) = Z. We’ll also have Hi(∂M) = 0 for i > 1. Similarly, M is homotopy equivalent to a circle, so we have H0(M) ∼= H1(M) ∼= Z and all others are 0. So far we have the long exact sequence
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تاریخ انتشار 2014