# Tagged Questions

17 views

### Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1}$ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n$. What is the ...
46 views

### Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
63 views

68 views

### Universal coefficient formula

Let $X$ be a compact manifold (so that all (co)homologyies have finite rank). The universal coefficient formula ($\mathbb{Z}$-coefficient for simplicity) says that we have the following short exact ...
1k views

### Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
396 views

### cohomology vs homology

I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for ...
581 views

### 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 ...
I am reading the Hatcher's book and am a bit confused with (co)homologies with coefficients. I would really appreciate if somebody clarifies the following to me. Let $C$ be a chain complex of ...