Subsection 3.3. (0N60) · ScalingStacks

3.3. Homology theories

We now give our second main definition of this paper, that of a homology theory. First, note that taking products of manifolds defines a functor $\mfld_{n-1}^{B}\times\mfld_{1}^{\sf or}\rightarrow\mfld_{n}^{B}$ , where $\mfld_{1}^{\sf or}$ is oriented 1-manifolds and

\mfld_{n-1}^{B}:=\mfld_{n-1}\underset{\spaces_{/{\sf BTop(n)}}}{\times}\spaces_{/B}

is the $\oo$ -category of $(n-1)$ -manifolds with a $B$ -framing on the product of their tangent bundle product with a trivial line bundle. Consequently, any $B$ -framed $n$ -manifold of the form $M_{0}\times\mathbb{R}$ , where $M_{0}$ is an $(n-1)$ -manifold, can be given the structure of a $\disk_{1}^{\sf or}$ -algebra in $\mfld_{n}^{B}$ , since $\mathbb{R}$ has the structure of a $\disk_{1}^{\sf or}$ -algebra in $\mfld_{1}^{\sf or}$ .

0N42

Definition 3.13 (Collar-gluing). A collar-gluing among $B$ -framed $n$ -manifolds is a continuous map

f\colon M\to[-1,1]

to the closed interval for which the restriction $f_{|}\colon M_{|(-1,1)}\to(-1,1)$ is a manifold bundle. We will often denote a collar-gluing $M\xrightarrow{f}[-1,1]$ simply as the open cover

M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}~\cong~M

where $M^{\prime}=f^{-1}[-1,1)$ and $M^{\prime\prime}=f^{-1}(-1,1]$ and $M_{0}=f^{-1}\{0\}$ .

0N43

Remark 3.14. We find it useful to think of a collar-gluing as the data of a manifold $M$ together with a codimension-1 properly embedded submanifold $M_{0}\subset M$ that splits the manifold $M$ into two disconnected parts, $M^{\prime}$ and $M^{\prime\prime}$ . Such data is afforded by gluing two manifolds with boundary along a common boundary. The actual data of a collar-gluing specifies that named just above, in addition to a bi-collaring $M_{0}\times\mathbb{R}\hookrightarrow M$ of $M_{0}\subset M$ .

Construction 2.21 offers, for each collar-gluing $M\xrightarrow{f}[-1,1]$ among $B$ -framed $n$ -manifolds, a monoidal functor

f^{-1}\colon\disk^{\partial,\sf or}_{1/[-1,1]}\longrightarrow\mfld^{B}_{n/M}~.

In particular, for each symmetric monoidal functor $\mfld^{B}_{n}\xrightarrow{\mathcal{F}}\mathcal{V}$ with $\otimes$ -presentable codomain, there is a canonical morphism in $\mathcal{V}$ :

(6)

\mathcal{F}(M^{\prime})\underset{\mathcal{F}(M_{0}\times\mathbb{R})}{\bigotimes}\mathcal{F}(M^{\prime\prime})~\underset{\rm Cor~\ref{interval}}{\simeq}~\int_{[-1,1]}\mathcal{F}\circ f^{-1}~\longrightarrow~\mathcal{F}(M)~.

0N44

Definition 3.15. A symmetric monoidal functor $\mathcal{F}:\mfld_{n}^{B}\rightarrow\mathcal{V}$ satisfies $\otimes$ -excision if, for each collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ among $B$ -framed $n$ -manifolds, the canonical morphism (6)

\mathcal{F}(M^{\prime})\underset{\mathcal{F}(M_{0}\times\mathbb{R})}{\bigotimes}\mathcal{F}(M^{\prime\prime})\xrightarrow{~\simeq~}\mathcal{F}(M)

is an equivalence in $\mathcal{V}$ . The $\oo$ -category of homology theories for $B$ -framed $n$ -manifolds valued in $\mathcal{V}$ is the full $\oo$ -subcategory

\mathbf{H}(\mfld_{n}^{B},\mathcal{V})~\subset~\Fun^{\otimes}(\mfld^{B}_{n},\mathcal{V})

consisting of those symmetric monoidal functors that satisfy $\otimes$ -excision.

0N45

Remark 3.16. The behavior of a homology theory with coefficients in $\mathcal{V}$ depends critically on the symmetric monoidal structure chosen on $\mathcal{V}$ . For instance, for the symmetric monoidal $\infty$ -category $(\m_{k},\oplus)$ of $k$ -modules over a fixed field $k$ with direct sum, a homology theory is forced to be ordinary homology with coefficients in $k$ ; while for $(\m_{k},\otimes)$ , a homology theories is typically not a homotopy invariant of manifolds.

0N46

Remark 3.17. One can complete the $\oo$ -category $\mfld_{n}$ as follows: first, formally adjoin, for every collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ , the colimit of the simplicial object ${\sf Bar}_{\bullet}(M^{\prime},M_{0}\times\mathbb{R},M^{\prime\prime})$ ; second, Dwyer-Kan localize by forcing the natural map from this new object $|{\sf Bar}_{\bullet}(M^{\prime},M_{0}\times\mathbb{R},M^{\prime\prime})|\rightarrow M$ to be an equivalence. Denote this completion of the $\oo$ -category of manifolds as $\widehat{\mfld}_{n}$ . The completion functor $\mfld_{n}\rightarrow\widehat{\mfld}_{n}$ is the universal homology theory: that is, we now have the suggestive equivalence

\int_{M}\mathbb{R}^{n}~\simeq~M

as objects of $\widehat{\mfld}_{n}$ (the lefthand side is not defined in $\mfld_{n}$ ). By this universal property, a $\otimes$ -excisive functor $\mfld_{n}\rightarrow\mathcal{V}$ is equivalent to a symmetric monoidal functor $\widehat{\mfld}_{n}\rightarrow\mathcal{V}$ that preserves geometric realizations of simplicial objects.

David Ayala, John Francis

Original source: arXiv:1206.5522v6