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It sounds as if you're confused about the difference between singular/simplicial homology and cellular homology. If you look at the homology chapter of Hatcher's book again, you'll see that these subjects are covered in some detail. The relevant details are as follows: The homology of a space is a deep property of that space that is hidden behind a ...

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I'll explain an analytic approach to computing the de Rham cohomology, to complement Qiaochu's algebraic one. Put any Riemannian metric on the torus, then there is a natural map from the space $\mathcal{H}^k(T^n)$ of harmonic $k$-forms to the de Rham cohomology $H^k_{dR}(T^n,\mathbb{R})$, where we send a harmonic form to its de Rham class. In fact, the Hodge ...

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$\mathbb{R}^n$ is contractible, so this description tells you that the torus is a classifying space $B \mathbb{Z}^n$, or equivalently an Eilenberg-MacLane space $K(\mathbb{Z}^n, 1)$, and hence its cohomology can be identified with the group cohomology of $\mathbb{Z}^n$. You can also compute this using the Kunneth formula, but there are other ways. The ...

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You may just apply Borsuk-Ulam theorem directly. Define a function $f$ from $S^n$ to $\mathbb{R}^n$ as follows: If $x$ is a point on $S^n$, then there is a great $S^{n-1}$ that is orthogonal to the point $x$. The $S^{n-1}$ divides $S^n$ into two regions. Let's call them $U$ and $V$, where $U$ is the region containing $x$. If $\mu$ is the measure given, ...

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(Posting an answer to get this off the unanswered queue.) Ralph Mellish mentions that proof of the kind requested can be found as Cor 1.26 in Vick's Homology Theory. I note the same proof appears as theorem 2.28 in Chapter 2.2 of Hatcher's Algebraic Topology.

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