5-manifolds: 1-connected

Revision as of 11:53, 1 September 2009

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1 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$
• $M_\infty \coloneq S^2 \times S^3$
• $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$
• $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$.

• Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a regular neighbourhood of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum.

• For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above.

• Reference for the Barden construction as twisted doubles.
• !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.

2 Invariants

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  $H_2(M)$ be the second integral homology group of

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  $M$, with torsion subgroup $TH_2(M)$.
• $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$.
• $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If

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  $M$ is Spin we set $h(M) = 0$.

• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$.

By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$.

For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$.

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3 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets.

Theorem 3.1 [Smale1962,]. There is a bijective correspondence

$$\displaystyle \mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 3.2 [Barden1965]. The mapping

$$\displaystyle \mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

This theorem can re-phrased in categorical language as follows: let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{W}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$.

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Theorem 3.4 [Barden1965]. The functor

$$\displaystyle (b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$$

is a detecting functor. That is, it is surjective on objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$.

3.1 Enumeration

We give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union

$$\displaystyle \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$

$$\displaystyle \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$

4 Further discussion

• As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.

• The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.

• By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$.

• As the invariants for $-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

4.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ and is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$

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• !!! Add references here.

4.2 Curvature and contact structures

• Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature [Boyer&Galicki2006].

• Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure [Thomas1986].

4.3 Mapping class groups

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we obtain an exact sequence

$$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$

where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$.

• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold.

• The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group [[[#|]]].

• In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.

5 References

• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Smale1962] S. Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903MediaWiki:Stub

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1 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$
• $M_\infty \coloneq S^2 \times S^3$
• $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$
• $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$.

• Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a regular neighbourhood of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum.

• For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above.

• Reference for the Barden construction as twisted doubles.
• !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.

2 Invariants

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  $H_2(M)$ be the second integral homology group of

  Tex syntax error

  $M$, with torsion subgroup $TH_2(M)$.
• $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

  Tex syntax error

  $M$, $w_2 \in H^2(M; \Zz_2)$.
• $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If

  Tex syntax error

  $M$ is Spin we set $h(M) = 0$.

• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

  Tex syntax error

  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$.

By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$.

For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$.

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3 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets.

Theorem 3.1 [Smale1962,]. There is a bijective correspondence

$$\displaystyle \mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 3.2 [Barden1965]. The mapping

$$\displaystyle \mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

This theorem can re-phrased in categorical language as follows: let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{W}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$.

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Theorem 3.4 [Barden1965]. The functor

$$\displaystyle (b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$$

is a detecting functor. That is, it is surjective on objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$.

3.1 Enumeration

We give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union

$$\displaystyle \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$

$$\displaystyle \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$

4 Further discussion

• As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.

• The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.

• By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$.

• As the invariants for $-M$ are isomorphic to the invariants of

  Tex syntax error

  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

4.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ and is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$

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• !!! Add references here.

4.2 Curvature and contact structures

• Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature [Boyer&Galicki2006].

• Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure [Thomas1986].

4.3 Mapping class groups

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we obtain an exact sequence

$$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$

where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$.

• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold.

• The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group [[[#|]]].

• In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

  Tex syntax error

  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.

5 References

• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Smale1962] S. Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903MediaWiki:Stub

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1 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$
• $M_\infty \coloneq S^2 \times S^3$
• $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$
• $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$.

• Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a regular neighbourhood of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum.

• For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above.

• Reference for the Barden construction as twisted doubles.
• !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.

2 Invariants

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  $H_2(M)$ be the second integral homology group of

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  $M$, with torsion subgroup $TH_2(M)$.
• $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$.
• $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If

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  $M$ is Spin we set $h(M) = 0$.

• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$.

By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$.

For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$.

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3 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets.

Theorem 3.1 [Smale1962,]. There is a bijective correspondence

$$\displaystyle \mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 3.2 [Barden1965]. The mapping

$$\displaystyle \mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

This theorem can re-phrased in categorical language as follows: let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{W}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$.

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Theorem 3.4 [Barden1965]. The functor

$$\displaystyle (b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$$

is a detecting functor. That is, it is surjective on objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$.

3.1 Enumeration

We give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union

$$\displaystyle \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$

$$\displaystyle \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$

4 Further discussion

• As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.

• The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.

• By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$.

• As the invariants for $-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

4.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ and is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$

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• !!! Add references here.

4.2 Curvature and contact structures

• Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature [Boyer&Galicki2006].

• Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure [Thomas1986].

4.3 Mapping class groups

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we obtain an exact sequence

$$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$

where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$.

• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold.

• The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group [[[#|]]].

• In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

  Tex syntax error

  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.

5 References

• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Smale1962] S. Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903MediaWiki:Stub

; Let $\pi_0\Diff_{+}(M)$ denote the group of [[isotopy]] classes of orientation preserving diffeomorphisms $f\co M \cong M$ and let $\Aut(H_2(M))$ be the group of isomorphisms of $H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification [[#Classification|above]] we obtain an exact sequence $$\mathcal{M}_{5}(e)$ be the set of diffeomorphism classes of closed, oriented, smooth, simply-connected 5-manifolds

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$M$ and let $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of spinable manifolds. In this article we report the calculation of $\mathcal{M}_5^{\Spin}(e)$ first obtained in [Smale1962] and of $\mathcal{M}_{5}(e)$ first obtained in general in [Barden1965].

Contents 1 Constructions and examples 2 Invariants 3 Classification 3.1 Enumeration 4 Further discussion 4.1 Bordism groups 4.2 Curvature and contact structures 4.3 Mapping class groups 5 References

1 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$
• $M_\infty \coloneq S^2 \times S^3$
• $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$
• $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$.

• Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a regular neighbourhood of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum.

• For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above.

• Reference for the Barden construction as twisted doubles.
• !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.

2 Invariants

Consider the following invariants of a closed simply-connected 5-manifold

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$M$.

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  $H_2(M)$ be the second integral homology group of

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  $M$, with torsion subgroup $TH_2(M)$.
• $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$.
• $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If

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  $M$ is Spin we set $h(M) = 0$.

• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$.

By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$.

For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$.

An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that

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$b(x, y) = -b(y, x)$ and

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$b(x, y) = 0$ for all

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$y \in H$ if and only if

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$x = 0$. By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$. Moreover $H$ must be isomorphic to

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$T \oplus T$ or

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$T \oplus T \oplus \Zz_2$ for some finite group $T$ with

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$b(x,x) = 1/2$ if

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$x$ generates the $\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold

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$M$ determines the isomorphism class of the linking form

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$b_M$ and we see that the torsion subgroup of

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$H_2(M)$ is of the form

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$TH_2(M) \cong T \oplus T$ if

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$h(M) \neq 1$ or

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$TH_2(M) \cong T \oplus T \oplus \Zz_2$ if

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$h(M) = 1$ in which case the $\Zz_2$ summand is an orthogonal summand of

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$b_M$.

3 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets.

Theorem 3.1 [Smale1962,]. There is a bijective correspondence

$$\displaystyle \mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 3.2 [Barden1965]. The mapping

$$\displaystyle \mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

Theorem 3.3 [Barden1965, Theorem 2.2]. Let $M_0$ and $M_1$ be simply-connected, closed, smooth 5-manifolds and let $A\co H_2(M_0) \cong H_2(M_1)$ be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then $A$ is realised by a diffeomorphism.

This theorem can re-phrased in categorical language as follows: let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{W}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$.

Now let $\widetilde{\mathcal{M}}_5(e)$ be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let $(b, w_2) \co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$ be the funtor which maps

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$M$ to $(H_2(M), b_M, w_2(M))$ and $f \co M_0 \cong M_1$ to the induced map on $H_2$.

Theorem 3.4 [Barden1965]. The functor

$$\displaystyle (b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$$

is a detecting functor. That is, it is surjective on objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$.

3.1 Enumeration

We give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union

$$\displaystyle \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$

where the last two sets denote the diffeomorphism classes of non-Spinable 5-manifolds which are respectively boundaries and not boundaries. Then

$$\displaystyle \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$

4 Further discussion

• As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.

• The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.

• By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$.

• As the invariants for $-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

4.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ and is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$

and $w_2(X_{-1}) \neq 0$. It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ and that a closed, smooth 5-manifold

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$M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ where $M_0$ is a Spin manifold.

• !!! Add references here.

4.2 Curvature and contact structures

• Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature [Boyer&Galicki2006].

• Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure [Thomas1986].

4.3 Mapping class groups

Let $\pi_0\Diff_{+}(M)$ denote the group of isotopy classes of orientation preserving diffeomorphisms $f\co M \cong M$ and let $\Aut(H_2(M))$ be the group of isomorphisms of

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$H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above

we obtain an exact sequence

$$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$

where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$.

• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold.

• The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group [[[#|]]].

• In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

  Tex syntax error

  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.

5 References

• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Smale1962] S. Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903MediaWiki:Stub

\rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$ where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$. * Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold. * The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group {{cite|}}. * In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of $M$, has been extensively investigated by {{cite|Baues&Buth1996}} and is already seen to be relatively complex.

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1 Constructions and examples

We first list some familiar 5-manifolds using Barden's notation:

• $X_0 \coloneq S^5$
• $M_\infty \coloneq S^2 \times S^3$
• $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$
• $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$.

• Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a regular neighbourhood of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum.

• For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and second Stiefel-Whitney class $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above.

• Reference for the Barden construction as twisted doubles.
• !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties.

2 Invariants

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  $H_2(M)$ be the second integral homology group of

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  $M$, with torsion subgroup $TH_2(M)$.
• $w_2 \co H_2(M) \rightarrow \Zz_2$, the homomorphism defined by evaluation with the second Stiefel-Whitney class of

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  $M$, $w_2 \in H^2(M; \Zz_2)$.
• $h(M) \in \Nn \cup \{\infty\}$, the smallest extended natural number $r$ such that $x^{2^r} = e$ and $x \in w_2^{-1}(1)$. If

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  $M$ is Spin we set $h(M) = 0$.

• $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the linking form of

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  $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$.

By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$.

For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$.

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3 Classification

We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets.

Theorem 3.1 [Smale1962,]. There is a bijective correspondence

$$\displaystyle \mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$

Theorem 3.2 [Barden1965]. The mapping

$$\displaystyle \mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) , \quad [M] \mapsto ([H_2(M)], h(M))$$

is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$.

The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.

This theorem can re-phrased in categorical language as follows: let $\mathcal{Q}_5(e)$ be a small category, in fact groupoid, with objects $(G, b, w)$ where $G$ is a finitely generated abelian group, $b \co TG \times TG \to \Qq/\Zz$ is a anti-symmetric non-singular linking form and $w\co G \to \Zz_2$ is a homomorphism such that $w(x) = b(x, x)$ for all $x \in TG$. The morphisms of $\mathcal{W}_5(e)$ are isomorphisms of abelian groups commuting with both $w$ and $b$.

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Theorem 3.4 [Barden1965]. The functor

$$\displaystyle (b, w_2)\co \widetilde{\mathcal{M}}_5(e) \to \mathcal{Q}_5(e)$$

is a detecting functor. That is, it is surjective on objects and $M_0 \cong M_1$ if and only if $(b, w_2)(M_0) \cong (b, w_2)(M_1)$.

3.1 Enumeration

We give Barden's enumeration of the set $\mathcal{M}^{}_5(e)$, [Barden1965, Theorem 2.3].

• $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$.
• For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed above.
• For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed above be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$.

With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by

$$\displaystyle X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$

where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the connected sum of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$.

An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union

$$\displaystyle \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$

$$\displaystyle \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$

4 Further discussion

• As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.

• The classification of simply-connected Spin 5-manifolds was one of the first applications of the h-cobordism theorem.

• By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$.

• As the invariants for $-M$ are isomorphic to the invariants of

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  $M$ we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.

4.1 Bordism groups

As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group $\Omega_5^{\Spin}$ vanishes.

The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ and is detected by the Stiefel-Whitney number $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups

$$\displaystyle H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$

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• !!! Add references here.

4.2 Curvature and contact structures

• Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature [Boyer&Galicki2006].

• Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a contact structure [Thomas1986].

4.3 Mapping class groups

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we obtain an exact sequence

$$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$

where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$.

• Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold.

• The isotopy group $\pi_0\Diff_{+}(S^5)$ is known to be the trivial group [[[#|]]].

• In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of

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  $M$, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.

5 References

• [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
• [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
• [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
• [Smale1962] S. Smale, On the structure of $5$-manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
• [Thomas1986] C. B. Thomas, Contact structures on $(n-1)$-connected $(2n+1)$-manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
• [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
• [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903MediaWiki:Stub