# Tagged Questions

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### $C_g \simeq SX$ and $C_h \simeq SY$ [on hold]

Hi need some help with this problem: Let $f : X \to Y$ . Then we can form the cofiber sequence $X \to Y \to C_f \to C_g \to C_h$ where $g: Y \to C_f$, $h: C_f \to C_g$, and $i: C_g \to C_h$. Show ...
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### (Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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### Obtaining Chain Complex from a Cochain Complex

In this question: Constructing a cochain complex out of a chain complex , palio asked how to construct a co-chain complex when given a chain complex as well as how to go in the opposite direction, ...
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### Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
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### Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
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### In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
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### Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
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### Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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### Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
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My teacher told me the Kadison's process(may be not this ward, it is just my translation ) can make a 2-cocycle turn to be a cocycle(i.e.,derivation). But I can not find it in the internet. Thanks a ...
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### cohomology of an exact sequence

$$0\to M\to Q_1\to Q_2\to\dots\to Q_i \to N\to 0$$ exact sequence, then $$H^n(N)\cong H^{n+i}(M)$$