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# 数学代写|代数拓扑代考Algebraic Topology代考|MATH6510 Interactions Between Loops and Destabilization

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## 数学代写|代数拓扑代考Algebraic Topology代考|Interactions Between Loops and Destabilization

Recall from Proposition 2.2.3.17 that, for $t \in \mathbb{N}$, there is a natural isomorphism between $\Omega^t D, D \Sigma^{-t}: \mathscr{M} \rightrightarrows \mathscr{U}$. The following result is another application of Proposition 2.3.2.1:

Corollary 2.3.3.1 For $M$ an $\mathscr{A}$-module, there is a natural short exact sequence:
$$0 \rightarrow \Omega\left(D_s M\right) \rightarrow D_s\left(\Sigma^{-1} M\right) \rightarrow \Omega_1\left(D_{s-1} M\right) \rightarrow 0$$
Proof Let $F_{\bullet} \rightarrow M$ be a free resolution of $M$ (in $\mathscr{M}$ ) and take $C_{\bullet}=D F_{\bullet}$, which is a complex of projective unstable modules by Proposition 2.2.3.13.

Proposition $2.2 .3 .17$ implies that $\Omega C_{\bullet}$ is naturally isomorphic to $D\left(\Sigma^{-1} F_{\bullet}\right)$; $\Sigma^{-1} F_{\bullet}$ is a projective resolution of $\Sigma^{-1} M$, hence the homology of $\Omega C_{\bullet}$ calculates the derived functors $D_s\left(\Sigma^{-1} M\right)$, whereas the homology of $C_{\bullet}$ calculates the derived functors $D_s M$. The result follows immediately from Proposition 2.3.2.1.

Remark 2.3.3.2 The module $\Omega_1\left(D_{s-1} M\right)$ is the obstruction to $\Omega_s\left(D_s M\right) \rightarrow$ $D_s\left(\Sigma^{-1} M\right)$ being an isomorphism. This is zero if and only if $D_{s-1} M$ is reduced, by Corollary 2.3.1.9.

Remark 2.3.3.3 For $m \in \mathbb{N}$ and an $\mathscr{A}$-module $M$, there is a Grothendieck spectral sequence
$$\Omega_p^m D_q M \Rightarrow D_{p+q} \Sigma^{-m} M .$$
The short exact sequence of Corollary 2.3.3.1 corresponds to the case $m=1$.

## 数学代写|代数拓扑代考Algebraic Topology代考|Connectivity for Ds

The explicit identification of the destabilization functor $D M=M / B M$ (see Exercise 2.2.3.10) leads to the following result:

Lemma 2.3.4.1 For $M$ an $\mathscr{A}$-module, the natural surjection $M \rightarrow D M$ is an isomorphism in degrees $\leq 2(\operatorname{conn} M+1)$.

Proof The lowest degree element (if it exists-i.e. if $\operatorname{conn}(M)$ is finite) of $M$ has degree conn $(M)+1$, hence the lowest degree element of $B M$ has degree at least $2(\operatorname{conn}(M)+1)+1$. The result follows.

The following statement is a general result for connected algebras, stated here for the Steenrod algebra.

Lemma 2.3.4.2 An $\mathscr{A}$-module $M$ has a free resolution $F_{\bullet} \rightarrow M$ in $\mathscr{M}$ with $\operatorname{conn}\left(F_s\right) \geq \operatorname{conn}(M)+s$.
Proof An exercise for the reader.

The following weak result is sufficient for the initial applications; a much stronger result holds (combine Lemma 2.5.1.6 with Theorem 2.5.1.8).
Proposition 2.3.4.3 For $0<s \in \mathbb{N}$ and $M$ an $\mathscr{A}$-module
$$\operatorname{conn}\left(D_s M\right) \geq 2(\operatorname{conn} M+s)$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Interactions Between Loops and Destabilization

$$0 \rightarrow \Omega\left(D_s M\right) \rightarrow D_s\left(\Sigma^{-1} M\right) \rightarrow \Omega_1\left(D_{s-1} M\right) \rightarrow 0$$

$$\Omega_p^m D_q M \Rightarrow D_{p+q} \Sigma^{-m} M .$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Connectivity for Ds

$$\operatorname{conn}\left(D_s M\right) \geq 2(\operatorname{conn} M+s)$$

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