# Tag Info

Accepted

### If a map induces the identity on homology, does it also on cohomology?

Let $X=S^3\vee\Sigma\mathbb{RP}^2$ and consider the following map $f:X\to X$. On the $S^3$ summand, $f$ is the identity. On the $\Sigma\mathbb{RP}^2$ summand, $f$ is the composition of the pinch map ...
• 332k
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### What is the relation between homotopy groups and homology?

As commented above the Hurewicz theorem computes some homology groups from the knowledge of homotopy groups; it indicates that homotopy groups are harder to compute than homology groups. In degree 1 ...
• 1,515
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### Question about isomorphism given by the Serre Spectral sequence

In the Serre spectral sequence the edge homomorphism $$H_p(X;M)\to E^\infty_{p,0}\subset E^2_{p,0}=H_p(Y;M)$$ is the map induced by $X\to Y$. If $F$ is acyclic then the Serre spectral sequence ...
• 3,844

• 4,048
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### How simple are simple spaces in regards to twisted coefficients?

A simple space has the property that $\pi_1$ acts trivially on all homotopy groups, not just the higher ones, in particular $\pi_1$ is abelian. Otherwise there are very obvious counterexamples to your ...
• 11.7k
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### hodge number of complex surfaces

Here are a class of counterexamples. One can just consider the ruled surfaces $F_n$. These surfaces are not isomorphic for different $n$. However they have the same Hodge diamonds. One reference is ...
• 1,700
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### If $M$ is compact connected non-orientable 3-manifold, then $H_1(M)$ is infinite.

This is simply false. Consider $M=RP^2 \times [0,1]$.
• 99.6k
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### Quotient of cohomology groups with different coefficents

Universal coefficients gives, for any abelian group $A$ and any $X$, an isomorphism $$H^1(X, A) \cong \text{Hom}(H_1(X), A)$$ (since $H_0(X)$ is free, so the Ext term vanishes). If in addition $H_1(X)$...
• 421k

### Degree of a map $\Bbb RP^n\to S^{n-1}\times S^1$

Here's another way. Any such degree $k$ map gives a composition $$S^n\to \mathbb{R}P^n\to S^{n-1}\times S^1.$$ The first map has degree $2$ when $n$ is odd, so the composition has degree $2k$. ...
• 1,274
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### A consequence of Hilbert 90

It turns out there really is only a $K^\times$'s worth of solutions: We can rewrite the key equation as $b=\sigma(b)c$. Assume that also $b' = \sigma(b')c$. Then $b' = bx$, for some $x \in L^\times$. ...
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### Twists of morphisms and torsors

To answer your direct question, it is useful to first abstract the situation slightly. That said, maybe the following will be helpful to help orient you to the rest of the answer. TL;DR: Implicitly ...
• 54.3k
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### Proof of the "Whitehead theorem for homology"

As noted in Weak Homotopy Equivalence Induces Isomorphisms on Homology, we can replace $Y$ by the mapping cylinder of $f$; the effect is that we can assume that $f$ is an inclusion. Then we have long ...
• 7,677

### Question about isomorphism given by the Serre Spectral sequence

$\require{AMScd}$ I think you can use the fact that the Serre spectral sequence converges naturally; here is a rough argument: Consider the following commutative diagram of fibrations \begin{CD} F @&...
• 2,351
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• 77.4k
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### Relationship between Cohomology groups/rings related to a topological group

By considering $G=S^1,$ so that $BG$ is the infinite-dimensional complex projective space, you can see immediately that there’s no easy relationship between your second two cohomologies. There is ...
• 52.6k
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### $p$-torsion in fundamental group of Lie group.

First, the fundamental group of a connected Lie group is abelian and finitely generated. Secondly, any finitely generated abelian group is the fundamental group of a connected Lie group. To see the ...
• 132k
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### Examples of CW-complexes wich are 1-acyclic but no simply conected.

The one general thing that can be said is simply to state the 1-dimensional Hurewicz theorem: $$H_1(X) \approx \pi_1(X) \, / \, [\pi_1(X),\pi_1(X)]$$ In words, $H_1(X)$ is isomorphic to the quotient ...
• 122k
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### Vanishing of $r$-fold cup-products

Notice that the cup product defines a map $$H^*(X,A) \times H^*(X,B) \longmapsto H^*(X,A\cup B).$$ If $U$ is a set in your cover (you need only assume that the inclusion $U\to X$ is nullhomotopic, not ...
• 122k
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### Degree of a map $\Bbb RP^n\to S^{n-1}\times S^1$

Here's yet another way to see that the degree of a map $f: \mathbb{R}P^n \to S^{n-1} \times S^1$ is $0$ when $n \geq 2$. For $n \geq 2$, we have $\pi_1(\mathbb{R}P^n) \cong \mathbb{Z}_2$ and \pi_1( ...
• 2,652
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### What if $X$ is not path connected?

Intuitively, it could not possibly be true for a non path-connected spaces in general. Notice one object possibly depends on $x_0$ (in this case) whereas the other object is invariant; it does not ...
• 41.5k
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### Defining Atiyah-Hirzebruch spectral sequence from exact couple

The $(-1,1)$ indicates the degree, i.e. $\alpha$ maps the summand $D^{p,q} = h^{p+q}(X_p)$ into $h^{p+q}(X_{p-1})=D^{p-1,q+1}.$ Analogously for the other maps.
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### Homology groups of $X \times S^1$

Let us understand $S^1$ as the set of complex numbers $z$ with $\lvert z \rvert = 1$. In Example 2.48 Hatcher derives an exact sequence "which is somewhat similar to the Mayer–Vietoris sequence ...
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### Proof of excission Axiom for singular Homology

In general, to say a chain $c\in\Delta_p(X)$ lies in a subspace $A\subseteq X$ just means it is contained in the subcomplex $\Delta_p(A)\subseteq\Delta_p(X)$, equivalently its support (the union of ...
• 12.4k
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### Is homology dual to cohomology or is homotopy dual to cohomology?

Homotopy and cohomology are dual in a vague sense called "Eckmann-Hilton duality". As the linked page says, this is more a heuristic rather than a precise notion of duality. Certainly in the ...
• 7,677

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