22
votes

Accepted

### If the universal cover of a manifold is spin, must it admit a finite cover which is spin?

No, this is not true: for each dimension $d \geq 4$, there is a closed, oriented $d$-manifold which is not spin, whose universal cover is spin, but which does not have a finite cover that is spin.
The ...

19
votes

Accepted

### Is a 4-dimensional submanifold of a spin manifold always spin?

Let $i$ denote an immersion $N \to M$. There is an exact sequence of vector bundles on $N$ given by
$$0 \to TN \to i^*TM \to \nu \to 0$$
where $\nu$ is the normal bundle. As total Stiefel-Whitney ...

17
votes

### Does Spin cobordism vanish in dimension $4k-1$?

I believe the bordism groups are nonzero in every dimension after some relatively small finite dimension, just by looking at the Poincaré polynomial in Anderson-Brown-Peterson's earlier paper "Spin ...

16
votes

### Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions

The condition for having a $Pin^+$-structure is the vanishing of $w_2$, and for having a $Pin^-$-structure is the vanishing of $w_2 +w_1^2$, for manifolds of any dimension. This is because the Lie ...

16
votes

### Topological obstruction for the existence of spin$^c$ structure

The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let
$$
\beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z})
$$
be the Bockstein ...

16
votes

Accepted

### Obstruction of spin-c structure and the generalized Wu manifods

Define the Wu manifold $W(n) = SU(n)/SO(n)$, the inclusion $SO \to SU$ given by thinking of $\Bbb C^n = \Bbb R^n \otimes \Bbb C$ (that is, including real matrices into complex matrices). Note that $W(...

14
votes

Accepted

### Spin^c structures on manifolds with almost complex structure

Assume that $M$ is oriented throughout. Recall that $M$ has a $\text{Spin}^c$ structure iff the third integral Stiefel-Whitney class $\beta w_2 = W_3 \in H^3(M, \mathbb{Z})$ is trivial. Actually more ...

14
votes

Accepted

### A corollary of the non-existence of positive scalar curvature

I am suspicious of your result.
The three torus $\mathbb{T}^3$ is well-known to not admit any metric of positive scalar curvature.
Let $g_0$ be the flat metric on $\mathbb{T}^3$. Given a positive ...

13
votes

### Spin-H structures

As mentioned in Arun Debray's answer, a closed orientable smooth manifold $M$ is spin${}^h$ if and only if there is a principal $SO(3)$-bundle (or equivalently, an orientable real rank three bundle) $...

12
votes

### Harmonic spinors on closed hyperbolic manifolds

I'm happy to be able to answer my own question! John Ratcliffe, Steven Tschantz and I showed that the Dirac operator on the Davis manifold (a closed hyperbolic 4-manifold constructed by Mike Davis) ...

11
votes

### Open questions in "Spin geometry"

Spin geometry is an active field and of course is not exhausted in the book of Lawson and Michelson.
In fact, nowadays, there are new books on the topic, including more recent results.
In the ...

11
votes

### Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions

For $d \geq 4$, let $M_d = (S^1)^{d-4}\times\mathbb{CP}^2$. As tori are parallelisable, $w(M_d) = w(\mathbb{CP}^2)$, in particular $w_1(M_d) = 0$ and $w_2(M_d) \neq 0$, so $M_d$ does not admit a Spin, ...

11
votes

### What is the relationship between spinors and supermanifolds and fermions?

You've got the right concepts, but they're presented in a way that makes me think some context could be helpful.
In #1, you're really talking about the special case where $V$ is one of the spinor ...

11
votes

Accepted

### The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?

You can work out the answers to these questions using the material in Chapter 11 of the book Spinors and Calibrations by F. Reese Harvey. You will also need to recall that, for $N\not=4$, the group ...

11
votes

### A question about the existence of spin maps

If $M$ and $N$ are spin, then every map between them is a spin map. In particular, there exist spin maps $M \to N$ of degree zero.
If $M$ is spin and $N$ is not spin, then $f : M \to N$ is a spin map ...

10
votes

### Action of the spin covariant derivative on gamma matrices?

The question is not clearly stated, but if appropriately interpreted it does make sense.
Let $S$ be a spinor bundle over a pseudo-Riemannian manifold $(M,g)$, whose bundle of Clifford algebras is ...

10
votes

### Spin-H structures

SpinH-structures were studied by Shiozaki-Shapourian-Gomi-Ryu for applications to condensed-matter physics. They prove in Lemma D.9 that a closed manifold $M$ admits a spinH-structure iff it's ...

10
votes

### Is a spin structure on a knot complement the same thing as an orientation of the knot?

The unique spin structure on $S^3$ restricts to a spin structure on $S^3 \setminus L$, and so provides a basepoint to the affine space of all spin structures on the latter.

9
votes

### Does Spin cobordism vanish in dimension $4k-1$?

The spin cobordism groups $\Omega^{spin}_n$ have been computed for $n \leq 127$; see section 10 of Secondary Invariants for String Bordism and tmf by Bunke and Naumann. They use MAPLE together with ...

8
votes

Accepted

### Open questions in "Spin geometry"

As @314159 has explained, Spin geometry on "spin manifolds" is a very active field of research, but the subject goes way beyond the domain spin manifolds. The key point to notice is that every pseudo-...

8
votes

### Spin^c structures on manifolds with almost complex structure

A $\text{Spin}^c$ structure is equivalent to (a homotopy class of) an almost complex structure on the 2-skeleton of a manifold which extends to the 3-skeleton (except for a surface or when the fiber ...

8
votes

Accepted

### Topological Spin manifolds in dimension 4

The map $\Omega_4^{\text{Spin}} \to \Omega_4^{\text{SpinTop}}$ is taken isomorphically by the signature to the inclusion $16\Bbb Z \hookrightarrow 8\Bbb Z$, so that the groups are abstractly ...

8
votes

Accepted

### Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$

A smooth manifold $M$ admits a pin$^+$ structure if and only if $w_2(M) = 0$, and a pin$^-$ structure if and only if $w_1(M)^2 + w_2(M) = 0$; see this page for some information on pin structures. The ...

7
votes

Accepted

### Recovering K-theory and KO-theory from KR-theory and Bott Periodicity Theory

Unless I am mistaken, $KU(X) = KR(X \times S^{0,1})$ for $X$ an ordinary space. To see why this is reasonable, unpack $X\times S^{0,1} = X\sqcup X$ with the obvious involution and check that Real ...

7
votes

Accepted

### Spin 4-manifold bounded by a mapping torus of tori

I think that there are three spin structure on T^2 that do not bound, and that we have three candidates spin structures on U that we don't know how to fill, a priori. (Thanks Danny for clarifying this....

7
votes

### Converse to Lichnerowicz Vanishing Theorem?

This is far from true! For a generic metric on a spin manifold of dimension at least 3, the kernel of the Dirac operator will be as small as it can be, subject to the index theorem. This was proved by ...

7
votes

Accepted

### Converse to Lichnerowicz Vanishing Theorem?

I think the answer is No. You are essentially asking the following: If $0$ is not an eigenvalue of the Dirac operator $D$ on a compact Riemannian manifold, then does the underlying Riemannian metric ...

7
votes

Accepted

### Explicit generators of the Lie algebra $spin(9)$

There are various places where you can see this written down, but let me suggest some notes that I wrote about spinors in the low dimensions that includes what you want, assuming that you know ...

7
votes

Accepted

### Spin Structure on AdS- Schwarzschild manifold

The existence or nonexistence of a spin structure on a smooth manifold $M$ is a topological question, in that it does not depend on the choice of a Riemannian metric on $M$. Spin structures are ...

7
votes

Accepted

### An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space

This is probably overkill, but I couldn't resist advertising a preprint that Diarmuid Crowley and I recently posted to the arXiv: https://arxiv.org/abs/1802.01296
In the final section we discuss ...

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