On the Number of Holes in a Common Drinking Straw

arXiv:2605.31831 [math.AT] · Preprint · v4 · 16 May 2026

M. R. Annulus¹ — Department of Beverage Topology, Institute for Applied Pedantry

Abstract. We settle, with what we concede is gratuitous machinery, the recurring popular question of how many holes are possessed by a standard drinking straw. The straw is homotopy equivalent to the circle; we therefore compute its homotopy groups, singular and de Rham cohomology rings, group (co)homology, complex and real K-theory, oriented bordism, stable homotopy, and an Alexander-dual linking invariant. For each theory we recall its definition and meaning, exhibit a short proof, and identify the resulting generator with the unique hole. Every invariant designed to count holes returns the same value, namely $1$. The two boundary circles occasionally mistaken for additional holes are demonstrated to be boundary, not topology.

Keywords. drinking straw · homotopy type · Bott periodicity · Alexander duality · beverage cylinder · Möbius distinction.

MSC 2020. 55-02, 55N15, 55N20, 55Q10, 57R75.

§ 1. Introduction

It is a recurring scandal of online life that the question how many holes does a drinking straw have? excites passionate disagreement; cf. [CV94] for a sober metaphysical treatment of holes as such. The proposed answers are typically one, two, or, in moments of zeal, zero. We resolve the matter using essentially every tool a graduate algebraic topology course leaves lying around [Hat02].

The strategy is elementary. A straw is homeomorphic to the cylinder $S^1\times[0,1]$, which deformation retracts onto its core circle $S^1$. Every homotopy-invariant feature of the straw therefore agrees with that of $S^1$. We compute these invariants in a representative selection of (co)homology theories. In each section we (i) recall the definition of the theory and what it measures, (ii) state the result, (iii) cite a standard reference, (iv) give a short proof, and (v) identify the generator with the unique hole.

Figure 1. Figure 1. A straw $X \simeq S^1\times[0,1]$ with its core circle $\gamma$ (dashed) representing the generator of $H_1(X;\mathbb{Z})$.

§ 2. A multi-theoretic census

Throughout, $X$ denotes a fixed straw.

Definition. A straw is a topological space $X$ homeomorphic to $S^1\times[0,1]$. The boundary circles $S^1\times\{0\}$ and $S^1\times\{1\}$ are called the mouth and the foot; the open subset $S^1\times(0,1)$ is the bore.

The projection $S^1\times[0,1]\to S^1$ is a deformation retraction, so $X\simeq S^1$. This single fact will do almost all of the work below.

§ 2.1. Homotopy groups

The homotopy groups $\pi_n(X,x_0)$ are defined as based-homotopy classes of maps $(S^n,*)\to(X,x_0)$, with group structure given by concatenation of loops (for $n=1$) or by stacking on cubical faces (for $n\geq 2$). They are the finest classical invariants we shall consider: $\pi_n$ records precisely which $n$-spheres can be mapped into $X$ up to deformation.

The circle is the prototypical Eilenberg–MacLane space $K(\mathbb{Z},1)$; the standard reference is Hatcher [Hat02, Thm 1.7 & §4.1]:

$$\begin{aligned} \pi_1(X) &= \mathbb{Z}, \\ \pi_n(X) &= 0 \quad (n \neq 1). \end{aligned}$$

Sketch. The map $p\colon\mathbb{R}\to S^1,\ t\mapsto e^{2\pi i t}$ is the universal covering, with fibre $\mathbb{Z}$. The path-lifting property gives a unique lift $\widetilde\ell$ of each loop $\ell\colon[0,1]\to S^1$ with $\widetilde\ell(0)=0$, and the difference $\widetilde\ell(1)-\widetilde\ell(0)\in\mathbb{Z}$ — the winding number — depends only on the homotopy class of $\ell$. The resulting map $\pi_1(S^1)\to\mathbb{Z}$ is a bijection.

For $n\geq 2$, the covering map induces an isomorphism $\pi_n(\mathbb{R})\xrightarrow{\sim}\pi_n(S^1)$, and $\pi_n(\mathbb{R})=0$ because $\mathbb{R}$ is contractible. Hence $\pi_1$ has a single free generator (the loop $\gamma$ winding once through the bore) and all higher $\pi_n$ vanish — one hole’s worth of loops. ∎

§ 2.2. Singular homology

Singular homology is built from chains: formal $\mathbb{Z}$-linear combinations of continuous maps $\Delta^n\to X$, quotiented by the relation $\partial^2=0$. The resulting groups $H_n(X;\mathbb{Z})$ measure $n$-cycles that fail to bound — informally, $n$-dimensional voids that $X$ is unable to fill in.

With integer coefficients, see [Hat02, Ex. 2.23]:

$$\begin{aligned} H_0(X;\mathbb{Z}) &= \mathbb{Z}, \\ H_1(X;\mathbb{Z}) &= \mathbb{Z}, \\ H_n(X;\mathbb{Z}) &= 0 \quad (n \geq 2). \end{aligned}$$

Sketch. Equip $S^1$ with the minimal CW structure: one $0$-cell $e^0$ and one $1$-cell $e^1$ with both endpoints attached to $e^0$. The cellular chain complex is

$$0\longrightarrow \mathbb{Z}\langle e^1\rangle \xrightarrow{\;\partial\;} \mathbb{Z}\langle e^0\rangle\longrightarrow 0,$$

and $\partial e^1=e^0-e^0=0$ since the two attaching endpoints cancel with opposite orientations. The homology of this complex is $\mathbb{Z}$ in degrees $0$ and $1$ and zero elsewhere. The single $\mathbb{Z}$ in degree one is generated by the fundamental class $[\gamma]$ of the core circle — the unique non-bounding $1$-cycle — and this is the hole. ∎

Remark. The mouth and foot cobound the cylinder and are therefore homologous in $X$. They contribute jointly to the single generator of $H_1$, not separately. This is the principal source of the popular two-hole heresy.

§ 2.3. Cohomology ring

Cohomology is the dual construction to homology: cochains are $\mathbb{Z}$-linear functions on chains, and the coboundary is the formal adjoint of $\partial$. The cup product $\smile$ then makes $H^*(X;\mathbb{Z})$ into a graded-commutative ring, whose multiplicative structure records how lower-dimensional cohomology classes assemble into higher-dimensional ones.

As a graded ring, after [Hat02, Ex. 3.13],

$$H^*(X;\mathbb{Z}) \;\cong\; \mathbb{Z}[\alpha]\big/(\alpha^2),$$

the exterior algebra on a one-dimensional generator $\alpha$.

Sketch. Since $H_*(X;\mathbb{Z})$ is free, the Universal Coefficient Theorem gives $H^n(X;\mathbb{Z})\cong\mathrm{Hom}(H_n(X;\mathbb{Z}),\mathbb{Z})$, so $H^0=H^1=\mathbb{Z}$ and $H^n=0$ for $n\geq 2$. For the ring structure, the cup product $\smile\colon H^1\otimes H^1\to H^2$ lands in the zero group, hence $\alpha\cup\alpha=0$ for any $\alpha\in H^1$. Therefore $H^*$ is generated by the single class $\alpha$ dual to $[\gamma]$, with $\alpha^2=0$ — one generator, one hole, no further structure. ∎

§ 2.4. de Rham cohomology

For a smooth manifold $M$, de Rham cohomology is the quotient $H^k_{\mathrm{dR}}(M)=\{\text{closed }k\text{-forms}\}/\{\text{exact }k\text{-forms}\}$. It measures the obstruction to a closed differential form being globally exact, and by the de Rham theorem agrees with singular cohomology over $\mathbb{R}$.

Working on the smooth open straw $S^1\times\mathbb{R}$, after [BT82, §I.4],

$$\begin{aligned} H^0_{\mathrm{dR}}(X) &= \mathbb{R}, \\ H^1_{\mathrm{dR}}(X) &= \mathbb{R}\cdot[d\theta]. \end{aligned}$$

Sketch. The angular one-form $d\theta$ is globally well-defined on $S^1\cong\mathbb{R}/2\pi\mathbb{Z}$ even though $\theta$ is only a local coordinate, and it is closed because every $1$-form on a $1$-manifold is. It is not exact: if $d\theta=df$ for some smooth $f$, then by the fundamental theorem of calculus $\oint_\gamma d\theta=0$, contradicting the explicit computation $\oint_\gamma d\theta=2\pi$.

By the de Rham theorem, $H^*_{\mathrm{dR}}(X;\mathbb{R})\cong H^*(X;\mathbb{R})$, so $H^1_{\mathrm{dR}}$ has rank one and must be generated by $[d\theta]$. The class $[d\theta]$ is precisely the obstruction to a closed $1$-form on the straw being exact, and this obstruction is the hole. ∎

§ 2.5. Group (co)homology

The (co)homology of a discrete group $G$ may be defined either as the derived functors of the $G$-invariants functor on $G$-modules or, equivalently, as the singular (co)homology of the classifying space $BG=K(G,1)$. It records intrinsic algebraic data about $G$ — extensions, characteristic classes, central extensions — in topological form.

The circle is the classifying space $B\mathbb{Z}=K(\mathbb{Z},1)$ [Bro82, Ch. II], hence

$$\begin{aligned} H_*(X;\mathbb{Z}) &= H_*(\mathbb{Z};\mathbb{Z}), \\ H^*(X;\mathbb{Z}) &= H^*(\mathbb{Z};\mathbb{Z}). \end{aligned}$$

Sketch. The group $\mathbb{Z}$ admits the short free resolution

$$0\longrightarrow \mathbb{Z}[\mathbb{Z}] \xrightarrow{\,t-1\,} \mathbb{Z}[\mathbb{Z}] \xrightarrow{\;\varepsilon\;} \mathbb{Z}\longrightarrow 0$$

as $\mathbb{Z}[\mathbb{Z}]=\mathbb{Z}[t,t^{-1}]$-modules, where $\varepsilon$ is the augmentation. Applying $-\otimes_{\mathbb{Z}[\mathbb{Z}]}\mathbb{Z}$ collapses $t-1$ to the zero map, leaving $\mathbb{Z}\xrightarrow{0}\mathbb{Z}$ and so $H_0(\mathbb{Z};\mathbb{Z})=H_1(\mathbb{Z};\mathbb{Z})=\mathbb{Z}$, $H_n=0$ otherwise. Equivalently, $S^1=B\mathbb{Z}$, so the group (co)homology of $\mathbb{Z}$ is, by definition, the (co)homology of $S^1$. The hole of the straw is the rank-one freeness of the integers as a group. ∎

§ 2.6. Complex K-theory

Complex K-theory $K^*(X)$ is built from the Grothendieck group of stable isomorphism classes of complex vector bundles on $X$, extended to a $\mathbb{Z}/2$-graded cohomology theory by Bott periodicity ($K^{n+2}\cong K^n$). It records the same broad type of information as ordinary cohomology, but with linear data over $X$ in place of cycles, and is closer to the differential geometry of the space.

After [Ati67, §1.4] and [Bot59],

$$\begin{aligned} K^0(X) &= \mathbb{Z}, \\ \widetilde K^{\,0}(X) &= 0, \\ K^1(X) &= \mathbb{Z}. \end{aligned}$$

Sketch. A complex vector bundle on $S^1$ is determined by its clutching function, an element of $\pi_0\, GL_n(\mathbb{C})$; since $GL_n(\mathbb{C})$ deformation-retracts onto $U(n)$ and $\pi_0\,U(n)=0$ for all $n$, every complex bundle on $S^1$ is trivial. Hence $\widetilde K^{\,0}(X)=0$ and $K^0(X)=\mathbb{Z}$ records rank only.

For the odd part, $K^1(X)=[X,U]_*=\pi_1(U)=\mathbb{Z}$ by the Bott periodicity theorem [Bot59]; the generator is the clutching function $z\mapsto z$, which represents the same loop as the core circle $\gamma$. The single $\mathbb{Z}$ in $K^1$ is the hole, this time recorded as a one-parameter family of inequivalent automorphisms over the bore. ∎

§ 2.7. Real K-theory; or, the Möbius distinction

Real K-theory $KO^*(X)$ is the analogue of complex K-theory using real rather than complex vector bundles, with Bott periodicity of order $8$ rather than $2$. The richer periodicity makes $KO$ sensitive to orientation, spin structure, and other real-geometric features that are invisible to either ordinary cohomology or complex K-theory.

After [Hus94, §16] and [Bot59],

$$\widetilde{KO}^{\,0}(X) \;=\; \mathbb{Z}/2,$$

generated by the Möbius line bundle.

Sketch. Real $n$-plane bundles on $S^1$ are classified by their clutching function $S^0\to O(n)$, i.e. by $\pi_0\, O(n)=\mathbb{Z}/2$ — the orientation component. The trivial element gives the trivial bundle; the nontrivial element gives the (rank-$n$) Möbius bundle. Stabilising in $n$ yields $\widetilde{KO}^{\,0}(S^1)=\mathbb{Z}/2$. Equivalently, by Bott periodicity for $KO$ (period $8$, with $\pi_*KO=\mathbb{Z},\mathbb{Z}/2,\mathbb{Z}/2,0,\mathbb{Z},0,0,0$), one has $\widetilde{KO}^{\,0}(S^1)=KO^{-1}(\mathrm{pt})=\pi_1(KO)=\mathbb{Z}/2$.

The group is cyclic on one generator: the same hole, but with strictly more information attached than complex K-theory or singular cohomology can carry. In particular, the straw and the Möbius strip — homotopy-equivalent and so indistinguishable to $H_*$ and $K^*$ — represent the trivial and nontrivial classes respectively, certifying the cylinder under consideration as a bona fide untwisted straw. ∎

§ 2.8. Oriented bordism

Oriented bordism $\Omega_n^{SO}(X)$ consists of equivalence classes of maps $f\colon M^n\to X$ from closed oriented $n$-manifolds, where $f_0\sim f_1$ iff they jointly extend over an oriented $(n{+}1)$-manifold-with-boundary mapping to $X$. It is a generalised homology theory whose cycles are honest manifolds rather than formal sums of simplices.

The coefficients vanish, $\Omega_1^{SO}(\mathrm{pt})=0$ [Sto68, Ch. VI], but the Atiyah–Hirzebruch spectral sequence yields

$$\Omega_1^{SO}(X) \;=\; \mathbb{Z}.$$

Sketch. The Atiyah–Hirzebruch spectral sequence for the generalised homology theory $\Omega_*^{SO}$ has $E^2_{p,q}=H_p(X;\Omega_q^{SO}(\mathrm{pt}))\Rightarrow\Omega_{p+q}^{SO}(X)$ [Sto68, Ch. VI], [MS74, App. A]. With $\Omega_0^{SO}=\mathbb{Z}$ and $\Omega_1^{SO}=0$, the relevant entries are $E^2_{1,0}=H_1(S^1;\mathbb{Z})=\mathbb{Z}$ and $E^2_{0,1}=0$; no differentials are possible into or out of these positions in total degree $\leq 1$, so $\Omega_1^{SO}(X)=\mathbb{Z}$.

A geometric representative for the generator is the inclusion $\gamma\colon S^1\hookrightarrow X$ of the core circle; the bordism class of a map $f\colon M^1\to S^1$ from a closed oriented $1$-manifold is detected by its total degree, $\Omega_1^{SO}(S^1)\xrightarrow{\sim}\mathbb{Z}$. The single $\mathbb{Z}$ is the hole, this time witnessed by oriented manifolds mapping into the bore. ∎

§ 2.9. Stable homotopy

Stable homotopy is the homology theory associated to the sphere spectrum, defined by $\widetilde\pi_n^{\,s}(X)=\mathrm{colim}_k\, \pi_{n+k}(\Sigma^k X)$ — homotopy classes of maps, stabilised under suspension. It is the universal generalised homology theory: every other homology theory receives a unique natural transformation from it.

After [Ada74, §2] and [Rav86, Ch. 1],

$$\widetilde\pi_n^{\,s}(X) \;\cong\; \pi_{n-1}^{\,s},$$

so the stable invariants of a straw are a degree shift of the stable stems:

$$\begin{aligned} &\mathbb{Z}, \;\mathbb{Z}/2, \;\mathbb{Z}/2, \;\mathbb{Z}/24, \\ &0, \;0, \;\mathbb{Z}/2, \;\mathbb{Z}/240, \;\ldots \end{aligned}$$

Sketch. The suspension isomorphism $\widetilde\pi_n^{\,s}(\Sigma Y)\cong\widetilde\pi_{n-1}^{\,s}(Y)$ is the defining shift property of the sphere spectrum [Ada74, §3]. Since $S^1\cong \Sigma S^0$, we obtain $\widetilde\pi_n^{\,s}(S^1)\cong\widetilde\pi_{n-1}^{\,s}(S^0)=\pi_{n-1}^{\,s}$, the $(n{-}1)$th stable stem. The first values are classical computations via the Adams spectral sequence and the $J$-homomorphism: $\pi_0^s=\mathbb{Z}$ (degree), $\pi_1^s=\mathbb{Z}/2\langle\eta\rangle$, $\pi_2^s=\mathbb{Z}/2\langle\eta^2\rangle$, $\pi_3^s=\mathbb{Z}/24\langle\nu\rangle$, $\pi_4^s=\pi_5^s=0$, $\pi_6^s=\mathbb{Z}/2\langle\nu^2\rangle$, $\pi_7^s=\mathbb{Z}/240\langle\sigma\rangle$ [Rav86, §1.1].

The hole shows up cleanly in degree $1$: $\widetilde\pi_1^{\,s}(X)=\pi_0^s=\mathbb{Z}$, generated by the inclusion of $\gamma$. Every other class is a fact about the geometry of exotic spheres which, by an accident of stable algebra, also pertains to the straw. ∎

§ 2.10. Alexander duality, and why the hands agree

Alexander duality is not itself a (co)homology theory but a theorem relating the homology of a compact, locally contractible subspace $K\subset S^n$ to the cohomology of its complement: $\widetilde H_i(S^n\!\setminus\! K)\cong\widetilde H^{n-i-1}(K)$. The pairing is implemented geometrically by the linking number, which counts how many times a cycle in the complement winds around a cycle in $K$.

Embed the core $\gamma\colon S^1 \hookrightarrow S^3$. By Alexander duality [Hat02, Thm 3.44], [Mun84, §74],

$$\widetilde H_1\big(S^3\setminus S^1\big) \;\cong\; \widetilde H^{\,1}(S^1) \;=\; \mathbb{Z},$$

and the linking pairing

$$H_1(X) \;\otimes\; H_1(S^3\!\setminus\!X) \;\longrightarrow\; \mathbb{Z}$$

is the identity.

Sketch. For a compact, locally contractible $K\subset S^n$, Alexander duality asserts $\widetilde H_i(S^n\!\setminus\! K;\mathbb{Z})\cong\widetilde H^{n-i-1}(K;\mathbb{Z})$ [Hat02, Thm 3.44]. Taking $K=S^1$ and $n=3$, $i=1$, gives $\widetilde H_1(S^3\setminus S^1)\cong\widetilde H^{\,1}(S^1)=\mathbb{Z}$. The duality is implemented by the linking-number pairing $\mathrm{lk}\colon H_1(K)\otimes H_1(S^3\setminus K)\to\mathbb{Z}$, which with one generator on each side is the identity matrix.

Translated to the kitchen: there is exactly one independent way to thread a closed loop through the straw, and its threading is detected by the linking number with the core circle $\gamma$. This is the invariant one computes physically by passing a piece of string through the bore — fingers and Alexander agree. ∎

§ 3. Conclusion

Theorem (Main Theorem). Every (co)homological invariant designed to count holes returns the same value when evaluated on a drinking straw, namely $\;1\;$.

The two-hole interpretation conflates boundary with topology: the straw has two boundary circles, but they cobound the cylinder and represent a single class in $H_1$. The zero-hole interpretation conflates contractibility with embeddedness in $\mathbb{R}^3$, an error we do not dignify further. We rest.

Funding Unfunded. The corresponding author paid for the straw out of pocket.

Data availability The straw used in this study is available from the corresponding author upon reasonable request and a self-addressed envelope.

Competing interests The author drinks primarily through straws.

Acknowledgements The original question was posed in a moot on the Bluesky social network [TKB26]; the author thanks the moot for the prompt and acknowledges that no part of this work is, strictly speaking, necessary.

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Footnotes

1. Correspondence to annulus@beverage-topology.example. The author is grateful to the editor for accepting the only paper they have ever written that begins with a straw. ↩