What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher’s book and Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$. For the graded module structure, it is obtained from the homology struture.

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems to me like the natural way of calculating something of this sort although I know that it is also possible to […]

I am currently reading through May’s “Algebraic Topology” and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using the canonical homeomorphism $(D^{n}, S^{n-1}) \simeq (D^{p} \times D^{q}, D^{p} \times S^{q-1} \cup S^{p-1} \times D^{q})$. However, I remember hearing that a product […]

I am working through Hatcher’s book and I am having trouble while understanding the CW-complex structure of Lens spaces. It is on page 145. He proves it constructing it in an inductive process. I upload two pictures: I think I understand all the previous ideas but I struggle with the text in yellow. I just […]

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration given by the family $E_i := p^{-1}(X_i)$. First recall the definition: Definition.A CW complex $X$ is a space $X$ which […]

Background A CW complex is a Hausdorff space and it is the union of its some of its subsets called cells, and cells are homeomorphic images in $X$ of some closed $k$-balls. The weak topology of a CW complex X is defined as the topology having the property that a subset of $X$ is closed […]

I don’t understand the construction of open nbds $N_\epsilon(A)$ of $A$ in a CW-complex $X$ given in page 522 of Hatcher’s Book. Since the book is available for free online, I’ll just copy the entire paragraph: Next we describe a convenient way of constructing open neighborhoods $N_\epsilon(A)$ of subsets $A$ of a CW complex $X$, […]

This is a followup to my question here. Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the inclusion $A \to X$. Is there a formula relating the […]

Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the inclusion $A \to X$. My question is, is $Y \cup_f X$ a CW complex with $Y$ as a […]

I am currently reading in Hatcher’s book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood $N_{\varepsilon}(A)$. You might also look at page 4 in this reference top of page 4 The construction is inductive […]

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