Degree Theory

Show that if \(f : S^n \to S^n\) has degree \(d\) then the induced map \(f^* : H^n(S^n;G) \to H^n(S^n;G)\) is multiplication by \(d\).

Prove or disprove:

1. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(f_d : T^2 \to S^2\).

2. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(g_d : S^2 \to T^2\).

   Here \(S^2\) is the 2-sphere and \(T^2 = S^1 \times S^1\) is the 2-dimensional torus.

Let \(S_g\) denote an orientable surface of genus \(g\). So for example \(S_0\) is the 2-sphere and \(S_{1}\) is the torus.

1. If \(g>0\) show that every map \(S_0\to S_g\) is null-homotopic.
2. If \(g>h>0\) show that every map \(S_h\to S_g\) has degree 0, but show by example that there exist maps that are not null-homotopic.

For \(n\ge 3\) let \(X=S^{n-1}\times S^1\) be the product of an \((n-1)\)-sphere and a circle and let \(Y\) be obtained by attaching an \(n\)-cell to \(X\) where the attaching map is a degree \(k\) from the boundary of the cell to the \((n-1)\)-sphere \(S^{n-1}\times \{p\}\subset X\) via a degree \(k\) map. Calculate the homology of \(Y\).

Show that the degree of every map \(\mathbb {C}P^2\to \mathbb {C}P^2\) of the complex projective plane to itself is nonnegative.

A map \(f:S^n\to S^n\) is said to be even if \(f(-x)=f(x)\) for every \(x\in S^n\). Show that even maps have even degree, and in fact that the degree must be 0 when \(n\) is even.

Let \(X\) be the cell complex obtained from the circle \(S^1\) by attaching two 2-cells \(e^2_2\) and \(e^2_3\) with attaching maps of degrees 2 and 3 respectively. Compute the fundamental group of \(X\). Carefully state any theorems you are using.

Let \(M=T^2\times S^2\), and suppose \(\omega \) is a 4-form on \(M\) such that \(\int _M\omega =7\).

1. Define a smooth map \(f:M\to M\) that has degree 2.
2. Define a 4-form \(\theta \) on \(M\) such that \(\int _M\theta =56\), using both \(f\) and \(\omega \) nontrivially.

Let \(f:\mathbb {R}P^3\to \mathbb {R}P^3\) be defined by \[ f([x_0:x_1:x_2:x_3])=[x_0^2:x_1^2:x_2^2:x_3^2]. \] Compute the degree and the Lefschetz number of \(f\).

Prove or disprove:

1. For every \(d\in \mathbb {Z}\) there is a degree \(d\) map \(f_d:T^2\to S^2\).
2. For every \(d\in \mathbb {Z}\) there is a degree \(d\) map \(g_d:S^2\to T^2\).

Find all connected covering spaces of \(\mathbb {R}P^2\vee S^1\) of degree less than three. Which of these coverings are regular?

Using winding numbers, show that there is a complex number \(z\) such that \[ z^2=e^{-|z|^2}+i e^{-|z|}. \]

Find a degree 2 map from \(S^1\times S^1\) to \(S^2\).

A. Prove the Brouwer Fixed Point Theorem using homology theory. That is, if \(f:B^n\to B^n\) is continuous, where \(B^n\) denotes the closed unit ball in \(\mathbb {R}^n\), then \(f\) has a fixed point.

B. Prove the Fundamental Theorem of Algebra using degree theory.

Show that every map \(f:\Sigma _2\to \Sigma _3\) has degree 0.

Let \(f:S^n\to S^n\) be a continuous map of spheres whose degree \(D(f)\ne 1\). Show that there is a point \(x\in S^n\) such that \(f(x)=a(x)\) where \(a\) is the antipodal map.

Let \(X_p=D^4\cup _f S^3\) be the space obtained by attaching a 4-cell to the 3-sphere by an attaching map \(f:\partial D^4\to S^3\) of degree \(p\).

1. Compute the singular homology of \(H_*(X_p;\mathbb {Z})\) and \(H_*(X_p\times X_q;\mathbb {Z})\) for any \((p,q)\) using the CW-complex cell structure.
2. Find \(H_*(X_p\times X_q;\mathbb {Z})\) using the Künneth Theorem.

True or false: Any closed differential form of degree \(\ge 1\) on \(\mathbb {R}P^2\times \mathbb {R}P^2\) is exact. Justify your answer.

Show that every continuous map \(f:S^n\to S^n\) (\(n\ge 1\)) of degree 2 has a fixed point.