# Tagged Questions

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### Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
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### Intersection form and poincaré duality

Let $M$ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
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### Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
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### uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
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### The Poincare Lemma for Compactly Supported Cohomology

I´m reading the proof of The Poincare Lemma for Compactly Supported Cohomology there is a part in the proof that said in the text book Bott and Tu: $d \pi_{\ast} = \pi_{\ast} d$ in other words, ...
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### The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
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### Request for companion of Mariano Suárez-Alvarez's proof.

Mariano Suárez-Alvarez's answer to Cohomology of projective plane seems very interesting. However, there are three pieces I could not stitch up for one of his proofs. Wonder if someone may help? ...
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### How calculate the De Rham cohomology group of $3$-torus: $T^3$?

How do I calculate the De Rham cohomology group of the $3$-torus $T^3$? Here $T^3=S^1 \times S^1 \times S^1$. Using the Mayer-Vietoris sequence, I can show that $\dim H_3(T^3)=\dim H_0(T^3)=1$. But ...
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### Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega$$ from $H^n_{DR}(X)$ to ...
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### Confusion on Cech cohomology

From Harvard math qualification exam, 1990. Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that \pi_{0}(B_{n}\cap B_{m})\le ...
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### Homology and cohomology: why does Poincaré duality fail for domains with boundary?

Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic. For domains with boundary, it's easy to construct examples where ...
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### Cohomology of complex projective plane

How can I compute Cohomology of complex projective plane $CP^2$? Any magic like the one here?
How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...